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.