Question

If the sum of the magnitude of two vectors is 18 and the magnitude of their resultant is 12. If the resultant is perpendicular to one of the vectors then what are the magnitudes of the two vectors?

Answer 1

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Answer 2

The magnitude of two vectors are 5 and 13.

Explanation:

Let A and B be two vectors whose magnitude sum is 18 and R is the resultant of them as 12. If the two vectors are placed such that the tail of B starts from the head of A. Then the resultant R is at right angled to A forms a right angled triangle as shown in figure below.

As we know that,

$A+B = 18$

So, the side B is 18 - A

According to Pythagoras theorem,

$\mathrm{B}^{2}=\mathrm{A}^{2}+\mathrm{R}^{2}$

$(18-A)^{2}=A^{2}+12^{2}$

$324-36 \mathrm{A}+\mathrm{A}^{2}=\mathrm{A}^{2}+144$

$324-144=36 A$

$36 A=180$

$A=\frac{180}{36}=5$

As A =5, then B will be,

$A+B=18$

$5+B=18$

$B=18-5=13$

Thus, the two vectors are of magnitude 5 and 13.