Question

If $|\bar{a}| = 3, |\bar{b}| = 4, |\bar{a} - \bar{b}| = 5$, then $|\bar{a} + \bar{b}| =$

• A. 9
• B. 25
• C. 5
• D. 4

Hint:

Observe that $|\bar{a}|^2 + |\bar{b}|^2 = |\bar{a} - \bar{b}|^2$. This is the Pythagorean theorem, which proves that the vectors $\bar{a}$ and $\bar{b}$ are orthogonal (perpendicular). For orthogonal vectors, the diagonals of the parallelogram are equal in length.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
Given the magnitudes of two vectors $\bar{a}$ and $\bar{b}$ and their difference, we need to determine the magnitude of their sum.

Step 2: Key Formula or Approach:
Use the parallelogram law of vector addition: $|\bar{a} + \bar{b}|^2 + |\bar{a} - \bar{b}|^2 = 2(|\bar{a}|^2 + |\bar{b}|^2)$.

Step 3: Detailed Explanation:
Substitute the given values into the identity: $|\bar{a} + \bar{b}|^2 + (5)^2 = 2((3)^2 + (4)^2)$
$|\bar{a} + \bar{b}|^2 + 25 = 2(9 + 16)$
$|\bar{a} + \bar{b}|^2 + 25 = 2(25) = 50$
$|\bar{a} + \bar{b}|^2 = 50 - 25 = 25$
Taking the square root: $|\bar{a} + \bar{b}| = 5$.

Step 4: Final Answer:
The magnitude $|\bar{a} + \bar{b}|$ is $5$, corresponding to option (C).