Answer to Question #129149 in Analytic Geometry for Quanda

Question

If two collinear vectors A and B are added, the resultant has a magnitude equal to 4.0. If B is subtracted from A, the resultant has a magnitude equal to 8.0. What is the magnitude of A?

Answer 1

Vectors  A and B are collinear: $B=kA.$

If two collinear vectors A and B are added

$|A+kA|=4$

If B is subtracted from A

$|A-kA|=8$

$\dfrac{|1-k|}{|1+k|}=\dfrac{8}{4}$

$|1-k|=2|1+k|, k\not=\pm1$

If $k>1$

$k-1=2+2k$ $k=-3, False$

If $-1< k<1$

$1-k=2+2k$ $k=-\frac{1}{3}, True$

If $k<-1$

$1-k=-2-2k$ $k=-3, True$

$|A||1+(-3)|=4$

$|A|=2$

The magnitude of A is 2.

$|A||1+(-\frac{1}{3})|=4$

$|A|=6$

The magnitude of A is 6.