Question

The directions cosines of the joins of the following pairs of vectors :
(i) $(6,3,2),(5,1,4)$ be $\frac{k}{3},\frac{m}{3},-\frac{m}{3}$

(ii) $(3,-4,7),(0,2,5)$ be $\frac{n}{7},\frac{l}{7},\frac{m}{7}$
Find $k+m+n+l$ ?

Answer 1

For $P\left(x_1{y}_1{z}_1\right)$ and $Q\left(x_2{y}_2{z}_2\right)$

Direction cosines are $\frac{x_2-x_1}{PQ},\frac{y_2-y_1}{PQ},\frac{z_2-z_1}{PQ}$

Now for $P(6,3,2),Q(5,1,4)$

$PQ=\sqrt{{(6-5)}^2+{(3-1)}^2+{(2-4)}^2}=\sqrt{1+4+4}=3$

And direction cosine $\frac{6-5}{3},\frac{3-1}{3},\frac{2-4}{3}=\frac{1}{3},\frac{2}{3},\frac{-2}{3}$

And for $A(3,-4,7),B(0,2,5)$

$AB=\sqrt{{(3-0)}^2+{(-4-2)}^2+{(7-2)}^2}=\sqrt{9+36+4}=\sqrt{49}=7$

Direction cosine is $\frac{3-0}{7},\frac{-4-2}{7},\frac{7-5}{7}=\frac{3}{7},\frac{-6}{7},\frac{2}{7}$

$\therefore k+m+n+l=1+2+3-6=0$