If ` vec a= hat i+ hat j+ hat k , vec b=4 hat i-2 hat j+3 hat k` and ` vec c= hat i-2 hat j+ hat k` , find a vector of magnitude 6 units which is parallel to the vector `2 vec a- vec b+3 vec cdot`

Explanation

A vector parallel to `\vec{x}` is `\frac{\vec{x}}{\|\vec{x}|`


`\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b} &=4 \hat{i}-2 \hat{j}+3 \hat{k}, \vec{c}=\hat{i}-2 \hat{j}+\hat{k}`


`2 \vec{a}-\vec{b}+3 \vec{c}= 2(\hat{i}+\hat{j}+\hat{k})-(4 \hat{i}-2 \hat{j}+3 \hat{k})`
`+3(\hat{i}-2 \hat{j}+\hat{k})`

= `\hat{i}-2 \hat{j}+2 \hat{k}`



A vector of magnitude 6 units parallel to the vector `2 \vec{a}-\vec{b}+3 \vec{c}`


`=\frac{\hat{i}-2 \hat{j}+2 \hat{k}}{|\hat{i}-2 \hat{j}+2 \hat{k}|} \cdot 6`


=`\frac{\hat{i}-2 \hat{j}+2 \hat{k}}{\sqrt{1^{2}+2^{2}+2^{2}}} \cdot 6=\frac{\hat{i}-2 \hat{j}+2 \hat{k}}{3} \times 6`


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