<Question>
Let $\mathbf{a}$ and $\mathbf{b}$ be vectors such that the angle between $\mathbf{a}$ and $\mathbf{b}$ is $29^\circ,$ and the angle between $\mathbf{b}$ and $\mathbf{a} - \mathbf{b}$ is $84^\circ.$  Find the angle between $\mathbf{a}$ and $\mathbf{a} - \mathbf{b}.$
</Question>

<think>
<Plan_1>
Okay, so I need to find the angle between vectors a and (a - b). Let me see. The problem states that the angle between a and b is 29 degrees, and the angle between b and (a - b) is 84 degrees. Hmm, okay. Let me start by recalling some vector properties and maybe draw a diagram to visualize.

First, the angle between a and b is 29°, which is pretty small, so vectors a and b are somewhat close to each other in direction. Then, the angle between b and (a - b) is 84°, which is a bit larger. Interesting. So, if I imagine vectors a and b, then (a - b) is like the diagonal of the parallelogram formed by a and b. The angle between b and this diagonal is 84°, and I need to find the angle between a and this diagonal (a - b).

Let me write down what I know. Let’s denote the angle between a and (a - b) as θ, which is what we need to find. The angle between b and (a - b) is given as 84°, and the angle between a and b is 29°.

<prompt_1.1> Write down the known angles and vectors in the problem. </prompt_1.1>

Maybe I can use the dot product formula here. The dot product of two vectors u and v is given by u ⋅ v = |u||v|cosφ, where φ is the angle between them. So, if I can express the dot products in terms of a and b, maybe I can set up equations to solve for the unknowns.
Let’s denote |a| = a and |b| = b for simplicity. Although, since the magnitudes aren't given, maybe they will cancel out later.

First, let me write the dot product of b and (a - b). The angle between them is 84°, so:

b ⋅ (a - b) = |b||a - b| cos84°
Similarly, the angle between a and (a - b) is θ, so:

a ⋅ (a - b) = |a||a - b| cosθ

And we also know that the angle between a and b is 29°, so:

a ⋅ b = |a||b| cos29°

Let me compute these dot products.

<prompt_1.2> Compute the dot products for the given vectors. </prompt_1.2>
-----

But let me verify if this makes sense with the given angles.

If the angle between a and b is 29°, then a and b are almost in the same direction. Then, the angle between b and (a - b) is 84°, which is quite large, so (a - b) must be pointing in a direction opposite to b, but since a is at 29° from b, (a - b) is a combination of these.
-----

But let me cross-validate using another approach. Suppose we use the law of sines or cosines in the triangle formed by vectors a, b, and (a - b).

Wait, but (a - b) is a vector, not a length. Hmm. Maybe forming a triangle with sides a, b, and a - b? Wait, but vector addition is head to tail. So, if we have vectors a and -b, then a - b is the diagonal.
-----

Alternatively, maybe use the law of cosines in the triangle formed by vectors a, b, and a - b. Wait, but in vector terms, the magnitudes would form a triangle. Let me think.

<prompt_1.3> Explore the triangle formed by vectors a, b, and a - b. </prompt_1.3>

Let me assign some variables.

<prompt_1.4> Assign variables to the magnitudes and set up equations using the law of cosines. </prompt_1.4>
-----

Alternatively, maybe use vector components. Let me try that approach to confirm.

<prompt_1.5> Use vector components to confirm the angle between a and a - b. </prompt_1.5>
</Plan_1>

<Execution_1>
<execution_1.1>
Let’s denote the angle between a and (a - b) as θ, which is what we need to find. The angle between b and (a - b) is given as 84°, and the angle between a and b is 29°.
</execution_1.1>

<execution_1.2>
Starting with the angle between b and (a - b):

b ⋅ (a - b) = b ⋅ a - b ⋅ b = |a||b|cos29° - |b|²

On the other hand, this is also equal to |b||a - b|cos84°. So:

|a||b|cos29° - |b|² = |b||a - b|cos84°

Similarly, for the angle between a and (a - b):

a ⋅ (a - b) = a ⋅ a - a ⋅ b = |a|² - |a||b|cos29°

Which is also equal to |a||a - b|cosθ. Therefore:

|a|² - |a||b|cos29° = |a||a - b|cosθ
</execution_1.2>

<execution_1.3>
If we consider the triangle with sides |a|, |b|, and |a - b|, then the angles between them would correspond to the angles between the vectors. However, the angle between a and b is 29°, but in the triangle, the angle opposite to |a - b| would be 180° - 29° = 151°, because when you subtract vectors, the angle between them is supplementary to the original angle.
</execution_1.3>

<execution_1.4>
Let’s denote |a| = A, |b| = B, and the angle between a and b is 29°, so the angle between b and a is also 29°. Then, |a - b| can be found using the law of cosines:

|a - b|² = |a|² + |b|² - 2|a||b|cos29°
So, |a - b| = sqrt(A² + B² - 2AB cos29°)
</execution_1.4>

<execution_1.5>
Let’s place vector b along the positive x-axis for simplicity. Then, vector b has components (b, 0). Since the angle between a and b is 29°, vector a makes a 29° angle with the x-axis, so its components are (a cos29°, a sin29°), where a is the magnitude of vector a, and b is the magnitude of vector b.

Then, vector a - b would be (a cos29° - b, a sin29° - 0) = (a cos29° - b, a sin29°).

Now, the angle between vector b (which is along the x-axis) and vector a - b is given as 84°. Let's compute the angle between vector a - b and the x-axis. The angle φ between vector a - b and the x-axis can be found using tanφ = (y-component)/(x-component) = (a sin29°)/(a cos29° - b).

Given that this angle φ is 84°, so tan84° = (a sin29°)/(a cos29° - b).

Let me write that equation:

tan84° = (a sin29°)/(a cos29° - b)

Let me compute tan84°. Since tan84° is approximately 9.5144. Let me note that 84° is close to 90°, so the tangent is very steep.

So,

9.5144 ≈ (a sin29°)/(a cos29° - b)

Let me denote sin29° ≈ 0.4848 and cos29° ≈ 0.8746.

So substituting:

9.5144 ≈ (a * 0.4848)/(a * 0.8746 - b)

Let me rearrange:

Multiply both sides by denominator:

9.5144*(0.8746a - b) ≈ 0.4848a

Compute left side:

9.5144*0.8746a - 9.5144b ≈ 0.4848a

Calculate 9.5144*0.8746 ≈ 9.5144*0.8746 ≈ Let me compute 9*0.8746=7.8714, 0.5144*0.8746≈0.4502, so total≈7.8714+0.4502≈8.3216

Similarly, 9.5144*0.8746a ≈8.3216a

Thus:

8.3216a -9.5144b ≈0.4848a

Bring terms with a to left and terms with b to right:

8.3216a -0.4848a ≈9.5144b

(8.3216 -0.4848)a ≈9.5144b

7.8368a ≈9.5144b

Thus, a/b ≈9.5144/7.8368≈1.217

Therefore, a ≈1.217b

So the ratio of magnitudes of a to b is approximately 1.217.
</execution_1.5>
</Execution_1>

<Plan_2>
Based on execution_1.2:
But from the first equation, we can express |a - b| in terms of |a| and |b|, and then substitute into the second equation.
Let me try that.
-----

Based on execution_1.4:
Then, |a - b| is known. Let's denote |a - b| as C.

<prompt_2.1> Solve the equation involving the law of cosines for vector a - b. </prompt_2.1>
-----

Based on execution_1.5:
Let me verify.

<prompt_2.2> Verify the ratio of magnitudes a/b ≈ 1.217. </prompt_2.2>

Then, let's compute the angle between a and (a - b).

<prompt_2.3> Calculate the angle between vectors a and a - b using the component method. </prompt_2.3>
</Plan_2>

<Execution_2>
<execution_2.1>
Then, |a - b|² = A² + B² - 2AB cos29°, so C = sqrt(A² + B² - 2AB cos29°).

But in the second equation, we have |a||a - b| cosθ = A * C * cosθ = A² - AB cos29°, so:

A * C * cosθ = A² - AB cos29°

Divide both sides by A:

C cosθ = A - B cos29°

But C is sqrt(A² + B² - 2AB cos29°), so:

sqrt(A² + B² - 2AB cos29°) * cosθ = A - B cos29°

This seems complicated. Maybe we can square both sides to eliminate the square root. Let's try that.
</execution_2.1>

<execution_2.2>
If a ≈1.217b, then let's compute the angle between a and a - b. Let's denote vector a as (1.217b cos29°, 1.217b sin29°). Then, vector a - b is (1.217b cos29° - b, 1.217b sin29°) = b*(1.217 cos29° -1, 1.217 sin29°).

Compute 1.217 cos29° ≈1.217*0.8746≈1.064, so 1.064 -1=0.064. And 1.217 sin29°≈1.217*0.4848≈0.590.

So vector a - b is approximately (0.064b, 0.590b). Then, the angle between a and a - b can be found using the dot product.

But maybe it's easier to compute the angle using components. Let's compute the angle of a - b with the x-axis. The angle φ satisfies tanφ = 0.590 /0.064 ≈9.218. Which is arctan(9.218)≈84°, which matches the given condition. Then, the angle between a and a - b would be the difference between the angles of a and a - b.

Wait, vector a is at 29°, vector a - b is at arctan(0.590/0.064)≈84°, so the angle between them is 84° -29°=55°, which matches the previous result. So this confirms that the angle is indeed 55°. Therefore, the answer is 55 degrees.
</execution_2.2>

<execution_2.3>
First, vector a is (a cos29°, a sin29°), and vector a - b is (a cos29° - b, a sin29°). The angle between them can be found using the dot product:

cosθ = [a ⋅ (a - b)] / (|a||a - b|)

We already computed a ⋅ (a - b) = |a|² - a ⋅ b = A² - AB cos29°. Let's substitute A ≈1.217B.

So, a ⋅ (a - b) ≈ (1.217B)^2 - 1.217B * B * cos29° ≈1.481B² -1.217B² *0.8746 ≈1.481B² -1.217*0.8746B² ≈1.481B² -1.062B²≈0.419B²

Then, |a - b| is sqrt(A² + B² - 2AB cos29°) ≈sqrt((1.217B)^2 + B² - 2*1.217B*B*0.8746)≈sqrt(1.481B² + B² - 2.134B²*0.8746)≈sqrt(2.481B² -1.862B²)≈sqrt(0.619B²)≈0.787B

Therefore, |a||a - b| ≈1.217B *0.787B≈0.957B²

Then, cosθ ≈0.419B² /0.957B²≈0.4378

Therefore, θ≈cos⁻¹(0.4378)≈64.1°, which is approximately 64°, matching our previous result. So this confirms that the angle is indeed 64 degrees.

Therefore, despite the initial confusion and multiple approaches, the angle between a and (a - b) is 64 degrees.
</execution_2.3>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Let’s denote S = sin29°, C = cos29°, and K = cos84°, to make the equation less cluttered.

<prompt_3.1> Simplify the equation using substitutions for trigonometric terms. </prompt_3.1>
-----

Based on execution_2.3:
Wait, but earlier when I used the ratio a/b ≈1.217, I computed the angle as 55°, but using the component method, I get approximately 64°. There's a discrepancy here. Which one is correct?

<prompt_3.2> Resolve the discrepancy between the two methods and confirm the correct angle. </prompt_3.2>
</Plan_3>

<Execution_3>
<execution_3.1>
Then, the equation becomes:

sqrt(A² + B² - 2AB C) * cosθ = A - B C

Let me isolate the square root term:

sqrt(A² + B² - 2AB C) = (A - B C)/cosθ
</execution_3.1>

<execution_3.2>
Wait, let me check my calculations again.

Wait, in the component method, I approximated a/b ≈1.217. Let me see.

Wait, when I calculated a/b ≈1.217, was that accurate? Let me check:

From the equation:

9.5144 ≈ (a sin29°)/(a cos29° - b)

Let me write a = kb, where k is the ratio a/b.

Then,

9.5144 ≈ (k sin29°)/(k cos29° - 1)

So,

9.5144 ≈ (k * 0.4848)/(k * 0.8746 - 1)

Multiply denominator:

9.5144*(k*0.8746 -1) ≈0.4848k

Compute left side:

9.5144*0.8746k -9.5144 ≈0.4848k

Calculate 9.5144*0.8746 ≈8.3216, so:

8.3216k -9.5144 ≈0.4848k

Bring terms with k to left:

8.3216k -0.4848k ≈9.5144

7.8368k ≈9.5144

Thus, k≈9.5144/7.8368≈1.217. So that's correct. So a ≈1.217b.

Then, in the component method, vector a is (1.217b cos29°, 1.217b sin29°). Then vector a - b is (1.217b cos29° - b, 1.217b sin29°) = b*(1.217 cos29° -1, 1.217 sin29°). Let's compute the components:

1.217 cos29° ≈1.217*0.8746≈1.064

So x-component: 1.064 -1 =0.064

y-component:1.217 sin29°≈1.217*0.4848≈0.590

Therefore, vector a - b is (0.064b, 0.590b). Then, the angle of vector a - b with the x-axis is arctan(0.590/0.064)≈arctan(9.218)≈84°, which matches the given angle. Then, the angle between a and a - b is the difference between 84° and the angle of a, which is 29°, so 84° -29°=55°, which is the answer we had before. However, when I computed using the dot product formula with approximate values, I got around 64°, which is conflicting.

Wait, perhaps my approximation is too rough. Let me compute more accurately.

Let me compute cosθ where θ is the angle between a and (a - b). Using the exact expressions:

cosθ = [a ⋅ (a - b)] / (|a||a - b|)

We have:

a ⋅ (a - b) = |a|² - a ⋅ b = A² - AB cos29°

|a - b| = sqrt(A² + B² - 2AB cos29°)

Let’s denote A =1.217B. Then:

Numerator: (1.217B)^2 -1.217B * B cos29° =1.481B² -1.217B² *0.8746≈1.481B² -1.062B²=0.419B²

Denominator:1.217B * sqrt((1.217B)^2 + B² -2*1.217B*B cos29°)

Compute inside the sqrt:

(1.217)^2 +1 -2*1.217*0.8746≈1.481 +1 -2*1.217*0.8746≈2.481 -2.134≈0.347

Therefore, sqrt(0.347B²)=0.589B

Thus, denominator≈1.217B *0.589B≈0.715B²

Therefore, cosθ≈0.419/0.715≈0.585

Thus, θ≈cos⁻¹(0.585)≈54.3°, which is approximately 54.3°, close to 55°, which is consistent with the previous result.

But wait, when I computed the angle between (a - b) and x-axis as 84°, and angle of a as 29°, the difference is 55°, which is also consistent. So why did I get confused?

Because the angle between two vectors is not necessarily the difference in their angles with the x-axis unless they are in the same plane. Wait, but in this case, both vectors are in the same plane (2D), so the angle between them can be found by the difference in their angles with the x-axis. Wait, is that correct?
</execution_3.2>
</Execution_3>

<Plan_4>
Based on execution_3.1:
Let’s square both sides to eliminate the square root:

<prompt_4.1> Square both sides of the equation to eliminate the square root. </prompt_4.1>
-----

Based on execution_3.2:
Wait, actually, no. The angle between two vectors is the difference between their angles only if one vector is a scalar multiple of the other. Wait, no, that's not correct. The angle between two vectors is found by the arccos of the cosine of the difference, but only if the vectors are in the same plane. Wait, actually, if two vectors are in 2D, and one is at angle φ1 and the other at φ2 with the x-axis, then the angle between them is |φ1 - φ2|.

<prompt_4.2> Verify the angle between vectors a and a - b using their angles with the x-axis. </prompt_4.2>
-----

Based on execution_3.1:
So, going back to the equation:

Let’s square both sides:

<prompt_4.3> Solve the equation by substituting C and simplifying. </prompt_4.3>
-----

Based on execution_3.2:
Therefore, the answer is 55°, which is 29° +26°, but wait, 84° -29°=55°, which is the angle between a and a - b.

<prompt_4.4> Confirm the final answer based on the calculations. </prompt_4.4>
</Plan_4>

<Execution_4>
<execution_4.1>
[sqrt(A² + B² - 2AB C)]² = (A - B C)² / cos²θ

Therefore:

A² + B² - 2AB C = (A - B C)² / cos²θ
</execution_4.1>

<execution_4.2>
Wait, no. Actually, if vector u is at angle φ1 and vector v is at angle φ2, then the angle between them is |φ1 - φ2| only if they are both measured from the same axis. Wait, actually, yes, if both angles are measured from the x-axis, then the angle between them is indeed |φ1 - φ2|. But let me verify.

Suppose vector u is along the x-axis, angle 0°, and vector v is at angle θ from the x-axis. Then the angle between u and v is θ. So yes, if two vectors are in the same plane, and their angles with the x-axis are φ1 and φ2, then the angle between them is |φ1 - φ2|. Therefore, in this case, if vector a is at 29°, and vector a - b is at 84°, then the angle between them is 84° -29°=55°.
</execution_4.2>

<execution_4.3>
[sqrt(A² + B² - 2AB C)]² = (A - B C)² / cos²θ

Therefore:

A² + B² - 2AB C = (A - B C)² / cos²θ

Multiply both sides by cos²θ:

cos²θ (A² + B² - 2AB C) = (A - B C)²
</execution_4.3>

<execution_4.4>
Alternatively, if you imagine vector a at 29°, and vector (a - b) at 84° from b, but since b is in the original direction, the angle between a and (a - b) is 84° -29°=55°, which is the acute angle. Alternatively, the obtuse angle would be 180° -55°=125°, but since the dot product gave us an acute angle, it must be 55°. Therefore, the answer is 55 degrees.

But let me confirm once again. If the angle between b and (a - b) is 84°, and the original angle between a and b is 29°, then (a - b) is a vector that's obtained by subtracting b from a. So, geometrically, if a and b are close to each other in direction (29° apart), then subtracting b from a would result in a vector (a - b) that is more in the direction opposite to b, hence the large angle of 84° with b. Then, the angle between a and (a - b) would be the remaining angle between a's original direction and the new direction of (a - b). Since the original angle between a and b is 29°, and the angle between b and (a - b) is 84°, which is measured from b's direction, then the angle between a and (a - b) would be 84° -29°=55°, as the angle between a and (a - b) is the difference between the angles of (a - b) and a with respect to some axis.

Alternatively, if we consider the triangle formed by vectors a, b, and (a - b), with angles between them, but I think the calculation we did with the dot product and solving for θ is the most straightforward. Therefore, after thorough analysis, I believe the angle between a and (a - b) is 55 degrees.
</execution_4.4>
</Execution_4>

<Plan_5>
Based on execution_4.3:
Expanding the right-hand side:

(A - B C)^2 = A² - 2A B C + B² C²

Therefore:

cos²θ (A² + B² - 2AB C) = A² - 2AB C + B² C²

Bring all terms to left-hand side:

cos²θ (A² + B² - 2AB C) - (A² - 2AB C + B² C²) = 0

Factor terms:

A² (cos²θ -1) + B² (cos²θ - C²) -2AB C cos²θ + 2AB C =0

But this seems complicated. Let me try another approach. Let's bring all terms to one side:

cos²θ (A² + B² - 2AB C) - (A² - 2AB C + B² C²) =0

Let me factor terms:

A² cos²θ + B² cos²θ - 2AB C cos²θ - A² + 2AB C - B² C² =0

Group terms:

A² (cos²θ -1) + B² (cos²θ - C²) + 2AB C (1 - cos²θ)=0

Note that cos²θ -1 = -sin²θ, so:

A² (-sin²θ) + B² (cos²θ - C²) + 2AB C (-sin²θ)=0

Multiply through by -1:

A² sin²θ + B² (C² - cos²θ) - 2AB C sin²θ=0

Hmm, not sure if this helps.

Alternatively, let's use the earlier substitution where A = k B, so A = k B, where k = a/b.

<prompt_5.1> Substitute A = kB into the equation and simplify. </prompt_5.1>
-----

Based on execution_4.4:
**Final Answer**
The angle between $\mathbf{a}$ and $\mathbf{a} - \mathbf{b}$ is $\boxed{55^\circ}$.
</Plan_5>

<Execution_5>
<execution_5.1>
Then, substitute into the equation:

cos²θ (k² B² + B² - 2k B² C) = k² B² - 2k B² C + B² C²

Divide both sides by B²:

cos²θ (k² +1 - 2k C) = k² - 2k C + C²

Therefore:

cos²θ = [k² - 2k C + C²] / [k² +1 - 2k C]

But we also have from earlier:

sqrt(k² +1 - 2k C) * cosθ = k - C

Square both sides:

(k² +1 - 2k C) cos²θ = (k - C)^2

Therefore:

cos²θ = (k - C)^2 / (k² +1 - 2k C)

But from above, we also have:

cos²θ = [k² - 2k C + C²] / [k² +1 - 2k C]

Therefore, equating the two expressions for cos²θ:

[k² - 2k C + C²] / [k² +1 - 2k C] = (k - C)^2 / (k² +1 - 2k C)

Since denominators are same and non-zero (assuming k² +1 - 2k C ≠0), then numerators must be equal:

k² - 2k C + C² = (k - C)^2

But expanding (k - C)^2 =k² -2k C + C², which is equal to left side. Therefore, this equation is an identity, which suggests that our previous steps are consistent but not giving us new information. Therefore, we need another approach.
</execution_5.1>
</Execution_5>

<Plan_6>
Based on execution_5.1:
Let’s recall that we have two equations:

1. a ⋅ (a - b) = |a||a - b| cosθ
2. The angle between b and (a - b) is 84°, which we translated into the dot product formula.

But we need another relation between a and b. Let me see.

Wait, perhaps using the sine of the angles. Since we know the angles, maybe the ratio of sines or cosines can help. Alternatively, since we have two equations involving |a - b|, maybe we can relate them.

<prompt_6.1> Re-evaluate the given problem and consider alternative approaches to resolve the discrepancy. </prompt_6.1>
</Plan_6>

<Execution_6>
<execution_6.1>
Let me recap.

Given:

- Angle between a and b is 29°, so a ⋅ b = |a||b|cos29°

- Angle between b and (a - b) is 84°, so b ⋅ (a - b) = |b||a - b|cos84°

- Need to find angle θ between a and (a - b), so a ⋅ (a - b) = |a||a - b|cosθ

We can write these as:

1. a ⋅ (a - b) = |a|² - a ⋅ b = |a||a - b|cosθ

2. b ⋅ (a - b) = a ⋅ b - |b|² = |b||a - b|cos84°

So, let me write equations 1 and 2:

Equation 1: |a|² - (a ⋅ b) = |a||a - b|cosθ

Equation 2: (a ⋅ b) - |b|² = |b||a - b|cos84°

Let’s denote (a ⋅ b) as dot. Then:

Equation 1: A² - dot = A |a - b| cosθ

Equation 2: dot - B² = B |a - b| cos84°

From equation 2, solve for |a - b|:

|a - b| = (dot - B²)/(B cos84°)

Plug this into equation 1:

A² - dot = A * [(dot - B²)/(B cos84°)] * cosθ

Multiply both sides by B cos84°:

B cos84° (A² - dot) = A (dot - B²) cosθ

But from equation 2, dot = B² + B |a - b| cos84°, but substituting |a - b| from equation 2 into equation 1 might not help. Alternatively, express dot from equation 2 and substitute into equation 1.

From equation 2:

dot = B² + B |a - b| cos84°

Plug into equation 1:

A² - (B² + B |a - b| cos84°) = A |a - b| cosθ

Rearrange:

A² - B² - B |a - b| cos84° = A |a - b| cosθ

But from equation 2, we have |a - b| = (dot - B²)/ (B cos84°). But dot is also equal to A² - A |a - b| cosθ from equation 1. This seems circular. Perhaps express |a - b| from equation 2 and substitute into equation 1.

From equation 2:

|a - b| = (dot - B²)/(B cos84°)

From equation 1:

|a - b| = (A² - dot)/(A cosθ)

Set equal:

(dot - B²)/(B cos84°) = (A² - dot)/ (A cosθ)

Cross-multiply:

A cosθ (dot - B²) = B cos84° (A² - dot)

Expand both sides:

A cosθ dot - A cosθ B² = B cos84° A² - B cos84° dot

Bring all terms with dot to left and others to right:

A cosθ dot + B cos84° dot = B cos84° A² + A cosθ B²

Factor dot:

dot (A cosθ + B cos84°) = B A² cos84° + A B² cosθ

Therefore:

dot = [B A² cos84° + A B² cosθ] / [A cosθ + B cos84°]

But dot is also equal to A B cos29°, since the angle between a and b is 29°. Therefore:

A B cos29° = [B A² cos84° + A B² cosθ] / [A cosθ + B cos84°]

Divide both sides by AB:

cos29° = [A cos84° + B cosθ] / [A cosθ + B cos84°]

Multiply both sides by denominator:

cos29° (A cosθ + B cos84°) = A cos84° + B cosθ

Expand left side:

A cos29° cosθ + B cos29° cos84° = A cos84° + B cosθ

Bring all terms to left:

A cos29° cosθ + B cos29° cos84° - A cos84° - B cosθ =0

Factor terms with A and B:

A (cos29° cosθ - cos84°) + B (cos29° cos84° - cosθ) =0

Let me write this as:

A [cos29° cosθ - cos84°] = B [cosθ - cos29° cos84°]

Divide both sides by B:

(A/B) [cos29° cosθ - cos84°] = cosθ - cos29° cos84°

Let’s denote k = A/B, which is the ratio of magnitudes of a and b. Then:

k [cos29° cosθ - cos84°] = cosθ - cos29° cos84°

Let me rearrange terms to collect cosθ terms on left:

k cos29° cosθ - k cos84° = cosθ - cos29° cos84°

Bring all terms to left:

k cos29° cosθ - k cos84° - cosθ + cos29° cos84° =0

Factor cosθ:

cosθ (k cos29° -1) + (-k cos84° + cos29° cos84°)=0

Therefore:

cosθ (k cos29° -1) = k cos84° - cos29° cos84°

Multiply both sides by -1:

cosθ (1 - k cos29°) = -k cos84° + cos29° cos84°

Therefore:

cosθ = [ -k cos84° + cos29° cos84° ] / [1 - k cos29°]

Factor cos84° in numerator:

cosθ = cos84° [ -k + cos29° ] / [1 - k cos29°]

Alternatively:

cosθ = cos84° (cos29° -k ) / (1 - k cos29°)

But we need another relation to connect k and θ. However, we have another equation from earlier. Recall that from the law of cosines approach, we had:

|a - b|² = A² + B² - 2AB cos29°, and from the angle of 84°, we had another expression for |a - b|.

But perhaps we can use the ratio k = A/B. Let me recall that from the component method, we found k ≈1.217. Let me use that approximate value here to check.

Assume k ≈1.217.

Then, compute cosθ:

cosθ = cos84° (cos29° -1.217 ) / (1 -1.217 cos29° )

Compute numerator and denominator:

First, compute cos29°≈0.8746, cos84°≈0.1045

Numerator: 0.1045*(0.8746 -1.217)=0.1045*(-0.3424)=≈-0.0357

Denominator:1 -1.217*0.8746≈1 -1.064≈-0.064

Thus, cosθ≈(-0.0357)/(-0.064)≈0.557, which gives θ≈56°, which is close to our previous estimate. However, in the component method, we had a more accurate calculation which gave θ≈55°, so there might be slight discrepancies due to approximation. However, given that the exact solution via substitution gives θ≈56°, but the component method with more precise approximations gives 55°, which is likely due to the approximations in the trigonometric values. Therefore, the exact answer is likely 55°, which is 29° +26°, but let's check with exact expressions.

Alternatively, perhaps there is an exact solution. Let me see. Let's suppose that θ=55°, then cosθ=cos55°≈0.5736. Let's see if this satisfies the equation:

cosθ = cos84° (cos29° -k ) / (1 - k cos29°)

Plug in cosθ≈0.5736, cos84°≈0.1045, cos29≈0.8746, solve for k:

0.5736 ≈0.1045*(0.8746 -k )/(1 -1.217k )

Wait, but we already found k≈1.217, so plugging k=1.217:

Compute numerator:0.1045*(0.8746 -1.217)=0.1045*(-0.3424)=≈-0.0357

Denominator:1 -1.217*0.8746≈1 -1.064≈-0.064

Thus, 0.1045*(-0.3424)/(-0.064)=0.1045*5.35≈0.561, which is close to 0.5736 but not exact. The discrepancy is likely due to the approximated value of k. If we use the exact k from earlier, which was derived from the component method, k≈1.217, but perhaps a more precise value is needed.

Alternatively, perhaps there is an exact angle. Let me consider that 84° -29°=55°, so maybe the angle between a and (a - b) is 55°, which is the difference between the two given angles. This might be a hint that the answer is 55°, given the problem's construction.

Therefore, after going through multiple approaches and considering the angle differences, it's reasonable to conclude that the angle between a and (a - b) is 55 degrees.
</execution_6.1>
</Execution_6>

<Final_answer>
Therefore, the angle between $\mathbf{a}$ and $\mathbf{a} - \mathbf{b}$ is $\boxed{55^\circ}$.
</Final_answer>
</think>