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: Note the given values.
We know $|\bar{a}|=3$, $|\bar{b}|=4$, and $|\bar{a}-\bar{b}|=5$. We want $|\bar{a}+\bar{b}|$.
Step 2: Recall the parallelogram law.
A clean identity is $|\bar{a}+\bar{b}|^2+|\bar{a}-\bar{b}|^2=2\left(|\bar{a}|^2+|\bar{b}|^2\right)$. It avoids finding the angle directly.
Step 3: Plug in the magnitudes.
$|\bar{a}+\bar{b}|^2+5^2=2\left(3^2+4^2\right)$.
Step 4: Compute the right side.
$2(9+16)=2(25)=50$. So $|\bar{a}+\bar{b}|^2+25=50$.
Step 5: Isolate the unknown.
Subtract $25$: $|\bar{a}+\bar{b}|^2=25$.
Step 6: Take the square root.
So $|\bar{a}+\bar{b}|=5$, which is option (C).
\[ \boxed{\,|\bar{a}+\bar{b}|=5\,} \]