Question

The sum of the magnitudes of two vectors $\vec{A$ and $\vec{B}$ is $8$ and magnitude of the resultant is $4$. If the resultant vector is perpendicular to any one vector, then the magnitudes of the two vectors $\vec{A}$ and $\vec{B}$ are}

• A. $3, 5$
• B. $2, 6$
• C. $4, 4$
• D. $1, 7$

Hint:

Use Pythagoras theorem when the vectors are perpendicular and their sum is given.

Answer 1

The Correct Option is A

Step 1: Vector addition.
Let magnitudes of vectors $\vec{A}$ and $\vec{B}$ be $a$ and $b$ respectively.
From the given information:
\[ a + b = 8 \] Resultant magnitude of perpendicular vectors:
\[ R = \sqrt{a^2 + b^2} = 4 \] \[ a^2 + b^2 = 16 \]
Step 2: Solving the system of equations.
We now solve the system:
\[ a + b = 8 \] \[ a^2 + b^2 = 16 \] By solving, we get $a = 3$ and $b = 5$.

Step 3: Conclusion.
The magnitudes of the two vectors are $3$ and $5$.