Three Dimension...

If in a right-angled triangle \(A B C\), hypotenuse \(A C=p\), then what is \(\overrightarrow{A B} \cdot \overrightarrow{A C}+\overrightarrow{B C} \cdot \overrightarrow{B A}+\overrightarrow{C A} \cdot \overrightarrow{C B}\) equal to:

The direction ratios of a line are \((1,-3,2)\). Its direction cosines will be:

If vector \(b\) is vector whose initial point divides the join of \(5 i\) and \(5 j\) in the ratio \(k: 1\) and terminal point is origin and |vector \(\mathrm{b} \mid \leq \sqrt{37}\), then the set of exhaustive values of \(\mathrm{k}\) is:

If \(\overrightarrow{ a }=4 \hat{ i }+6 \hat{ j }\) and \(\overrightarrow{ b }=3 \hat{ j }+4 \hat{ k }\), then the vector form of the component of \(\vec{a}\) along \(\vec{b}\) is:

Find the direction cosines of the vector \(7 \hat{\imath}+4 \hat{\jmath}-3 \hat{k}\).

Find the value of \(x\) for which \(x(\hat{i}+\hat{j}+\hat{k})\) is a unit vector.

If \(|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}|=|\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}|\), then angle between \(\overrightarrow{\mathrm{a}}\) and \(\overrightarrow{\mathrm{b}}\) is:

Find the direction cosines of the vector \(\hat{\imath}+2 \hat{\jmath}-\mathrm{k}\).

If \(\overrightarrow{ a }=\hat{ i }+2 \hat{ j }-3 \hat{ k }, \overrightarrow{ b }=3 \hat{ i }-\hat{ j }+2 \hat{ k }\) then the angle between \(\vec{a}+\vec{b}, \vec{a}-\vec{b}\) is:

If â and \(\mathrm{b}\) are two unit vectors, then the vector \((\hat{a}+6) \times(\hat{a} \times 6)\) is parallel to: