Okay, let's see. I need to find the angle between vector a and vector (a - b). The problem gives me the angle between a and b, which is 29 degrees, and the angle between b and (a - b), which is 84 degrees. Hmm, so maybe I can use some vector properties or trigonometry here.

First, I remember that the angle between two vectors can be found using the dot product formula. The formula is:

cosθ = (u · v) / (|u| |v|)

So, maybe I can set up equations using this formula for the given angles and then solve for the unknowns. Let's denote the magnitude of a as |a|, the magnitude of b as |b|, and the angle we need to find as φ between a and (a - b).

Given:
1. The angle between a and b is 29°, so:
   (a · b) = |a||b|cos29°

2. The angle between b and (a - b) is 84°, so:
   [b · (a - b)] = |b||a - b|cos84°

Our target is to find φ, the angle between a and (a - b), which would be:
   [a · (a - b)] = |a||a - b|cosφ

So, maybe if I can express [a · (a - b)] in terms of |a|, |b|, and the known angles, I can solve for φ.

Let me start by expanding the second equation. The dot product of b and (a - b) is:

b · (a - b) = b · a - b · b = (a · b) - |b|²

From the first equation, we know that (a · b) = |a||b|cos29°, so substituting that in:

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

This is equal to |b||a - b|cos84°, so:

|b|(|a|cos29° - |b|) = |b||a - b|cos84°

We can divide both sides by |b| (assuming |b| ≠ 0, which it should be since the angle is given):

|a|cos29° - |b| = |a - b|cos84°

That's one equation. Now, let's look at the target expression:

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

And this is equal to |a||a - b|cosφ

So:

|a|² - |a||b|cos29° = |a||a - b|cosφ

Divide both sides by |a| (assuming |a| ≠ 0):

|a| - |b|cos29° = |a - b|cosφ

From the previous equation, we have:

|a|cos29° - |b| = |a - b|cos84°

So, let's denote |a - b| as D. Then:

From equation 1:
|a|cos29° - |b| = D cos84°

From the target expression:
|a| - |b|cos29° = D cosφ

So, we have two equations:

1. |a|cos29° - |b| = D cos84°
2. |a| - |b|cos29° = D cosφ

Let me write equations 1 and 2 again:

Equation 1:
|a|cos29° - |b| = D cos84°

Equation 2:
|a| - |b|cos29° = D cosφ

Our unknowns here are |a|, |b|, and D, but we need to find φ. However, we have three variables and two equations. Hmm, maybe we can find a relationship between |a| and |b| from equation 1 and substitute into equation 2.

From equation 1, let's solve for D:

D = (|a|cos29° - |b|) / cos84°

Then substitute D into equation 2:

|a| - |b|cos29° = [(|a|cos29° - |b|) / cos84°] * cosφ

Multiply both sides by cos84° to eliminate the denominator:

[|a| - |b|cos29°] cos84° = (|a|cos29° - |b|) cosφ

Let me rearrange this:

(|a| - |b|cos29°) cos84° = (|a|cos29° - |b|) cosφ

So, let's compute the left-hand side (LHS) and right-hand side (RHS):

LHS = |a|cos84° - |b|cos29°cos84°

RHS = |a|cos29°cos84° - |b|cos84°

So, set LHS = RHS:

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

Bring all terms to the left side:

|a|cos84° - |b|cos29°cos84° - |a|cos29°cos84° + |b|cos84° = 0

Factor terms with |a| and |b|:

|a|(cos84° - cos29°cos84°) + |b|(-cos29°cos84° + cos84°) = 0

Factor further:

|a|cos84°(1 - cos29°) + |b|cos84°(-cos29° + 1) = 0

Wait, notice that (1 - cos29°) is a common factor. Let's factor that out:

cos84°(1 - cos29°)(|a| + |b|) = 0

But cos84° is not zero, and (1 - cos29°) is also not zero (since 29° is not 180°). Therefore, the only possibility is |a| + |b| = 0. But magnitudes can't be negative, so |a| + |b| = 0 implies |a| = |b| = 0. But that can't be the case because then the angles wouldn't make sense. Hmm, that suggests a mistake in my algebra somewhere.

Wait, let me check the steps again. Let's go back to equation 1 and equation 2.

Equation 1: |a|cos29° - |b| = D cos84°
Equation 2: |a| - |b|cos29° = D cosφ

From equation 1, D = (|a|cos29° - |b|)/cos84°

Substitute into equation 2:

|a| - |b|cos29° = [(|a|cos29° - |b|)/cos84°] cosφ

Multiply both sides by cos84°:

(|a| - |b|cos29°) cos84° = (|a|cos29° - |b|) cosφ

Now, expanding LHS:

|a|cos84° - |b|cos29°cos84°

RHS:

|a|cos29°cos84° - |b|cos84°

Bring RHS to LHS:

|a|cos84° - |b|cos29°cos84° - |a|cos29°cos84° + |b|cos84° = 0

Factor terms:

|a|(cos84° - cos29°cos84°) + |b|(-cos29°cos84° + cos84°) = 0

Factor cos84° from both terms:

cos84°[|a|(1 - cos29°) + |b|(1 - cos29°)] = 0

Wait, hold on. Let me see:

cos84°[ |a|(1 - cos29°) + |b|(1 - cos29°) ] = 0

Because:

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

Similarly, -cos29°cos84° + cos84° = cos84°(1 - cos29°)

Therefore, we have:

cos84°(1 - cos29°)(|a| + |b|) = 0

Again, same result. Since cos84° ≠ 0 and (1 - cos29°) ≠ 0, then |a| + |b| = 0. Which is impossible. Hmm, so this suggests that there's a contradiction in the problem statement? But the problem gives specific angles, so maybe I made a wrong assumption.

Wait, maybe I made a mistake in the setup. Let me check again.

The dot product formula says that the angle θ between vectors u and v is given by cosθ = (u · v)/(|u||v|). So, for the angle between b and (a - b), which is 84°, the dot product is:

b · (a - b) = |b||a - b|cos84°

Which is correct. Then expanding that gives:

a · b - |b|² = |b||a - b|cos84°

And since a · b = |a||b|cos29°, substituting:

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

Divide both sides by |b| (assuming |b| ≠ 0):

|a|cos29° - |b| = |a - b|cos84°

That's correct. Then moving on to the target expression:

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

Which is equal to |a||a - b|cosφ

So:

|a|² - |a||b|cos29° = |a||a - b|cosφ

Divide both sides by |a|:

|a| - |b|cos29° = |a - b|cosφ

Which is correct. Then substituting |a - b| from the previous equation:

|a - b| = (|a|cos29° - |b|)/cos84°

So:

|a| - |b|cos29° = [(|a|cos29° - |b|)/cos84°] cosφ

Multiply both sides by cos84°:

(|a| - |b|cos29°)cos84° = (|a|cos29° - |b|)cosφ

Which is the same as before. Then expanding:

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

Bringing all terms to the left:

|a|cos84° - |b|cos29°cos84° - |a|cos29°cos84° + |b|cos84° = 0

Factor:

|a|(cos84° - cos29°cos84°) + |b|(-cos29°cos84° + cos84°) = 0

Which factors to:

|a|cos84°(1 - cos29°) + |b|cos84°(1 - cos29°) = 0

So:

cos84°(1 - cos29°)(|a| + |b|) = 0

Which implies |a| + |b| = 0, which is impossible. Hmm. That suggests there's a contradiction, which shouldn't be the case. So maybe I made a mistake in the angle between vectors.

Wait, the angle between b and (a - b) is 84°, so the angle is acute? 84° is less than 90°, so the angle is acute. The angle between a and (a - b) is what we need. Let me check if there's another approach.

Alternatively, maybe using the law of cosines on the triangle formed by the vectors a, b, and (a - b). Let me visualize this. If we consider the vectors a, b, and a - b, they form a triangle. Wait, but the angle between a and b is 29°, but the angle between b and (a - b) is 84°. So, maybe we can use the law of cosines in the triangle with sides |a|, |b|, and |a - b|.

Law of cosines states that for a triangle with sides a, b, c, and angle opposite side c is γ, then c² = a² + b² - 2ab cosγ.

In our case, the sides are |a|, |b|, and |a - b|. The angle opposite |a - b| is the angle between a and b, which is 29°, but wait, no. Wait, the angle between a and b is 29°, but the angle opposite |a - b| would be the angle at the origin between the vectors a and b. Wait, maybe not. Let's clarify.

Wait, actually, if we have vectors a, b, and a - b, the triangle is formed by the tips of the vectors. So, if you start at the origin, go to a, then from a to a - b, which is equivalent to going from a to a - b, which is the vector -b. So, the triangle would have sides |a|, |b|, and |a - b|, with angles between them as per the original problem.

But the angle between a and b is 29°, which is the angle at the origin between vectors a and b. The angle between b and (a - b) is 84°, which is the angle at the origin between vectors b and (a - b). Wait, but (a - b) is a vector starting at a and going to b? Wait, maybe not. Wait, in the vector diagram, the angle between b and (a - b) is the angle at the origin between the vectors b and (a - b). So, the angle between b and (a - b) is 84°, which is the angle between vectors b and (a - b). So, in the triangle formed by vectors a, b, and a - b, the angle at the origin between b and (a - b) is 84°, and between a and b is 29°, and the angle at the origin between a and (a - b) is what we need to find, which is φ.

So, using the law of cosines in this triangle:

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

But we also have another expression for |a - b| from the angle of 84°. Let's see.

Alternatively, we can use the law of sines. The law of sines states that in a triangle, the ratio of each side to the sine of the opposite angle is equal. So, if we denote the angles at the origin between the vectors as follows:

- Angle between a and b: 29° (opposite side |a - b|)
- Angle between b and (a - b): 84° (opposite side |a|)
- Angle between (a - b) and a: φ (opposite side |b|)

Wait, maybe not. Let me think again.

In the triangle formed by vectors a, b, and a - b, the sides are |a|, |b|, and |a - b|. The angles at the origin are:

- Between a and b: 29°
- Between b and (a - b): 84°
- Between (a - b) and a: φ

But in a triangle, the sum of the angles should be 180°. So, 29° + 84° + φ = 180°, so φ = 180° - 29° - 84° = 67°. Wait, is that possible?

Wait, that seems too straightforward. So, if the angles at the origin are 29°, 84°, and φ, then their sum must be 180°, so φ = 67°. But is that accurate?

Wait, but in reality, the triangle formed by the vectors a, b, and a - b is not a triangle with angles at the origin, but rather the vectors form a triangle when connected tip-to-tail. Wait, maybe I need to clarify.

If you have vectors a, b, and a - b, the triangle is actually the parallelogram formed by a and b, but the sides are a, b, and a - b. Wait, no. Wait, if you have vectors a and b, then a - b is the diagonal from b to a in the parallelogram. So, the triangle formed by the vectors a, b, and a - b is actually the triangle with sides |a|, |b|, and |a - b|, with angles at the origin between them. So, the angle between a and b is 29°, between b and (a - b) is 84°, and between (a - b) and a is φ.

Therefore, in the triangle, the three angles at the origin (between the vectors) are 29°, 84°, and φ. Since the sum of angles in a triangle is 180°, we have:

29° + 84° + φ = 180° ⇒ φ = 67°

Therefore, the angle between a and (a - b) is 67°. But wait, that seems too easy. Let me check if this makes sense.

Alternatively, let's verify using the law of cosines.

From the angle between a and (a - b), which is φ, we can use the law of cosines:

|a - b|² = |a|² + |a - b|² - 2|a||a - b|cosφ

Wait, no, that's not right. Wait, let's think again.

Wait, if we have the triangle with sides |a|, |b|, |a - b|, and angles at the origin 29°, 84°, and φ, then according to the law of sines:

|a| / sin84° = |b| / sinφ = |a - b| / sin29°

But we also have from the law of cosines:

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

But if we use the law of sines, we can write:

|a| / sin84° = |b| / sinφ

So, |a| = |b| sin84° / sinφ

Similarly, |a - b| / sin29° = |a| / sin84° ⇒ |a - b| = |a| sin29° / sin84°

But then substituting |a| from the first equation into the second:

|a - b| = (|b| sin84° / sinφ) * sin29° / sin84° = |b| sin29° / sinφ

So, |a - b| = |b| sin29° / sinφ

But we also have from the law of cosines:

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

Substituting |a| and |a - b| from above:

(|b| sin29° / sinφ)² = (|b| sin84° / sinφ)² + |b|² - 2*(|b| sin84° / sinφ)*|b|*cos29°

Divide both sides by |b|²:

(sin²29° / sin²φ) = (sin²84° / sin²φ) + 1 - 2*(sin84° / sinφ)*cos29°

Multiply both sides by sin²φ:

sin²29° = sin²84° + sin²φ - 2 sin84° cos29° sinφ

This seems complicated, but maybe we can use the angle φ = 67° as found earlier and check if it satisfies this equation.

Let me compute each term with φ = 67°:

sin²29° ≈ sin²(29) ≈ (0.4848)^2 ≈ 0.235

sin²84° ≈ sin²(84) ≈ (0.9945)^2 ≈ 0.989

sin²67° ≈ sin²(67) ≈ (0.9205)^2 ≈ 0.847

Now, compute the RHS:

sin²84° + sin²67° - 2 sin84° cos29° sin67°

= 0.989 + 0.847 - 2*0.9945*0.8746*0.9205

First, compute 2*0.9945*0.8746*0.9205:

≈ 2*0.9945 ≈ 1.989

1.989*0.8746 ≈ 1.741

1.741*0.9205 ≈ 1.600

So, RHS ≈ 0.989 + 0.847 - 1.600 ≈ 1.836 - 1.600 ≈ 0.236

Which is approximately equal to sin²29° ≈ 0.235. Close enough, considering rounding errors. Therefore, φ = 67° satisfies the equation. Therefore, the angle between a and (a - b) is 67°. So, the answer is 67 degrees.

But wait, let me confirm again. The initial approach with the triangle angles adding up to 180° seems to give the correct answer, and the verification via the law of sines and cosines also supports it. Therefore, the angle is 67 degrees.

**Final Answer**
\boxed{67^\circ}
Therefore, the angle between $\mathbf{a}$ and $\mathbf{a} - \mathbf{b}$ is $\boxed{67^\circ}$.
</think>