A vector ` vec r` has length 21 and its direction ratios are proportional to `2,-3,6.` Find the direction cosines and components of ` vec r` , is given that ` vec r` Makes an acute angle with `x-` axis.

Explanation

Let vector `vec{r}=a hat{i}+b hat{j}+c hat{k}`

Given `|vec{r}|=21`

and `frac{a}{2}=frac{b}{3}=frac{c}{-6}=lambda`
`therefore vec{r}=lambda(2 hat{i}+3 hat{j}-6 hat{k})
` and `|vec{r}|=21 `

`therefore 4 lambda^{2}+9 lambda^{2}+36 lambda^{2}=441`

`Rightarrow 49 lambda^{2}=441`

`Rightarrow lambda^{2}=9`

`Rightarrow lambda=3`

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