Question

Let $P=(1,0,-1)Q=(-1,2,0)R=(2,0,-3)S=(3,-2,-1)$, then the length of the component of $RS$ on $PQ$ is

• A. $1/3$
• B. $2/3$
• C. $4/3$
• D. $5/3$

Answer 1

Answer: (C)

$4/3$

$PQ(-1-1,2-0,0-(-1\left)\right);RS(3-2,-2-0,-1-(-3\left)\right)$

$PQ(-2,2,1);RS(1,-2,2)$

$\left|PQ\right|=3\left(\sqrt{4+4+1}\right);\left|RS\right|=3(\sqrt{1+{(-2)}^2+{\left(2\right)}^2}=\sqrt{9})$

$\cos \theta =\frac{(-2)\left(1\right)+\left(2\right)(-2)+\left(1\right)\left(2\right)}{\sqrt{{(-2)}^2+{\left(2\right)}^2+{\left(1\right)}^2}\sqrt{{\left(1\right)}^2+{(-2)}^2+{\left(2\right)}^2}}[\because \cos \theta =\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}]$

$\cos \theta =\frac{-4}{9}$

$\left|RS\right|\cos \theta =3\times \frac{4}{9}=\frac{4}{3}$ ($\because$ component of $RS$ on $PQ$ is $RS\cos \theta$)