Question

If $|\vec{a}| = 4$, $|\vec{b}| = 5$, then the values of $k$ for which $\vec{a} + k\vec{b}$ is perpendicular to $\vec{a} - k\vec{b}$ are

• A. $\pm \frac{5}{4}$
• B. $\pm \frac{2}{5}$
• C. $\pm \frac{16}{25}$
• D. $\pm \frac{4}{5}$

Hint:

Think of this problem geometrically: the vectors $(\vec{a} + k\vec{b})$ and $(\vec{a} - k\vec{b})$ represent the diagonals of a parallelogram formed by sides $\vec{a}$ and $k\vec{b}$. A parallelogram has perpendicular diagonals if and only if it is a rhombus (all sides equal). Therefore, set the side lengths equal: $|\vec{a}| = |k\vec{b}| \implies 4 = |k| \cdot 5 \implies |k| = \frac{4}{5} \implies k = \pm \frac{4}{5}$.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We are given the magnitudes of two vectors, $|\vec{a}| = 4$ and $|\vec{b}| = 5$. We need to find the scalar values of $k$ that make the linear combination vectors $(\vec{a} + k\vec{b})$ and $(\vec{a} - k\vec{b})$ perpendicular to each other.

Step 2: Key Formula or Approach:
Two non-zero vectors are perpendicular if and only if their vector dot product is exactly zero: $$(\vec{a} + k\vec{b}) \cdot (\vec{a} - k\vec{b}) = 0$$ We will use the distributive properties of the dot product along with the identity $\vec{v} \cdot \vec{v} = |\vec{v}|^2$.

Step 3: Detailed Explanation:
Expand the dot product expression using algebraic distribution: $$\vec{a} \cdot \vec{a} - k(\vec{a} \cdot \vec{b}) + k(\vec{b} \cdot \vec{a}) - k^2(\vec{b} \cdot \vec{b}) = 0$$ Since the dot product is commutative ($\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$), the middle cross-terms cancel each other out completely: $$|\vec{a}|^2 - k^2|\vec{b}|^2 = 0$$ Substitute the given numerical magnitudes ($|\vec{a}| = 4$ and $|\vec{b}| = 5$) into this equation: $$(4)^2 - k^2(5)^2 = 0$$ $$16 - 25k^2 = 0 \implies 25k^2 = 16$$ Isolate the squared scalar variable: $$k^2 = \frac{16}{25}$$ Taking the square root of both sides gives the final values for $k$: $$k = \pm \frac{4}{5}$$ This matches the choice provided in option (D).

Step 4: Final Answer:
The values of $k$ are $\pm \frac{4}{5}$, which corresponds to option (D).