© All Rights reserved @ LearnWithDash
Step-by-Step Solution
Step 1: Use the perpendicularity conditions
Given that
(\overrightarrow{a} + 3\overrightarrow{b}) \perp (7\overrightarrow{a} - 5\overrightarrow{b}),
it follows that their dot product is zero:
(\overrightarrow{a} + 3\overrightarrow{b}) \cdot (7\overrightarrow{a} - 5\overrightarrow{b}) = 0.
Likewise, from
(\overrightarrow{a} - 4\overrightarrow{b}) \perp (7\overrightarrow{a} - 2\overrightarrow{b}),
we get:
(\overrightarrow{a} - 4\overrightarrow{b}) \cdot (7\overrightarrow{a} - 2\overrightarrow{b}) = 0.
Step 2: Expand each dot product
Expand the first dot product:
(\overrightarrow{a} + 3\overrightarrow{b}) \cdot (7\overrightarrow{a} - 5\overrightarrow{b})
= 7|\overrightarrow{a}|^2 - 15|\overrightarrow{b}|^2 + (3 \cdot -5 + 1 \cdot 7)\,\overrightarrow{a}\cdot\overrightarrow{b}.
Carefully combining like terms, it simplifies to:
7|\overrightarrow{a}|^2 - 15|\overrightarrow{b}|^2 + 16(\overrightarrow{a} \cdot \overrightarrow{b}) = 0.
Label this equation as (1).
Now expand the second dot product:
(\overrightarrow{a} - 4\overrightarrow{b}) \cdot (7\overrightarrow{a} - 2\overrightarrow{b})
= 7|\overrightarrow{a}|^2 + 8|\overrightarrow{b}|^2 - 30(\overrightarrow{a} \cdot \overrightarrow{b}) = 0.
Label this as equation (2).
Step 3: Eliminate the dot product term
Multiply equation (1) by 30 and equation (2) by 16 to make the coefficient of \overrightarrow{a}\cdot \overrightarrow{b} the same (but opposite in sign).
Equation (1) × 30 gives:
210|\overrightarrow{a}|^2 - 450|\overrightarrow{b}|^2 + 480(\overrightarrow{a}\cdot\overrightarrow{b}) = 0.
Label this as equation (3).
Equation (2) × 16 gives:
112|\overrightarrow{a}|^2 + 128|\overrightarrow{b}|^2 - 480(\overrightarrow{a}\cdot\overrightarrow{b}) = 0.
Label this as equation (4).
Step 4: Add the resulting equations
Adding equations (3) and (4) to eliminate the dot product term, we get:
(210|\overrightarrow{a}|^2 + 112|\overrightarrow{a}|^2) + (-450|\overrightarrow{b}|^2 + 128|\overrightarrow{b}|^2) + (480 - 480)(\overrightarrow{a}\cdot\overrightarrow{b}) = 0.
Which simplifies to:
322|\overrightarrow{a}|^2 - 322|\overrightarrow{b}|^2 = 0 \quad \Rightarrow \quad |\overrightarrow{a}|^2 = |\overrightarrow{b}|^2.
Therefore,
|\overrightarrow{a}| = |\overrightarrow{b}|.
Step 5: Substitute into one of the original equations
From equation (2):
7|\overrightarrow{a}|^2 + 8|\overrightarrow{b}|^2 - 30(\overrightarrow{a} \cdot \overrightarrow{b}) = 0.
Since |\overrightarrow{a}| = |\overrightarrow{b}| , let |\overrightarrow{a}| = |\overrightarrow{b}| = r.
Then |\overrightarrow{a}|^2 = |\overrightarrow{b}|^2 = r^2. Also write \overrightarrow{a}\cdot \overrightarrow{b} = r^2 \cos\theta.
Substituting into equation (2), we get:
7r^2 + 8r^2 - 30(r^2 \cos\theta) = 0,
15r^2 = 30r^2 \cos\theta.
Dividing both sides by 15r^2 (assuming r \neq 0 ), we get:
1 = 2 \cos\theta \quad \Rightarrow \quad \cos\theta = \frac{1}{2}.
Step 6: Find the angle between the vectors
Since \cos\theta = \tfrac{1}{2}, the angle between \overrightarrow{a} and \overrightarrow{b} is:
\theta = 60^\circ.
