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If vec a ,\ vec b ,\ vec c are any ti...

If ` vec a ,\ vec b ,\ vec c` are any time mutually perpendicular vectors of equal magnitude `a ,` then `| vec a+ vec b+ vec c|` is equal to `a` b. `sqrt(2)a` c. `sqrt(3)a` d. `2a` e. none of these

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Since `vec{a}, vec{b} and vec{c} `are mutually perpendicular, so `vec{a} cdot vec{b}=vec{b} cdot vec{c}=vec{c} cdot vec{a}=0`
Angle between `vec{a} and vec{a}+vec{b}+vec{c}` is given by
`cos theta=frac{vec{a}cdot(vec{a}+vec{b}+vec{c})}{|vec{a}||vec{a}+vec{b}+vec{c}|}=frac{a^{2}}{|vec{a}||vec{a}+vec{b}+vec{c}|}`
Now we know, `|vec{a}|=|vec{b}|=|vec{c}|=a`
`|vec{a}+vec{b}+vec{c}|^{2}=|vec{a}|^{2}+|vec{b}|^{2}+|vec{c}|^{2}+2|vec{a}| cdot|vec{b}|+2|vec{b}| cdot|vec{c}|+2|vec{c}| cdot|vec{a}| `
`=a^{2}+a^{2}+a^{2}+0+0+0 `
`Rightarrow|vec{a}+vec{b}+vec{c}|=sqrt{3} a`
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