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The respective values of \(|\vec{a}|\) and \(|\vec{b}|\), if given \((\vec{a}-\vec{b})\cdot(\vec{a}+\vec{b})=512\) and \(|\vec{a}|=3|\vec{b}|\), are:

A student tries to tie ropes, parallel to each other from one end of the wall to the other. If one rope is along the vector \(3\hat{i}+15\hat{j}+6\hat{k}\) and the other is along the vector \(2\hat{i}+10\hat{j}+\lambda\hat{k}\), then the value of \(\lambda\) is :

The vectors \(\vec{a}=2\hat{i}-\hat{j}+\hat{k}\), \(\vec{b}=\hat{i}-3\hat{j}-5\hat{k}\) and \(\vec{c}=-3\hat{i}+4\hat{j}+4\hat{k}\) represents the sides of

Let \(\vec{a}\) be any vector such that \(|\vec{a}|=a\) The value of \(|\vec{a}\times\hat{i}|^{2}+|\vec{a}\times\hat{j}|^{2}+|\vec{a}\times\hat{k}|^{2}\) is:

The position vectors of points P and Q are \(\vec{p}\) and \(\vec{q}\) respectively. The point R divides line segment PQ in the ratio 3:1 and S is the mid-point of line segment PR. The position vector of S is: