Three Dimension...

What is the sum of the squares of direction cosines of the line joining the points \((-2,3,1)\) and \((1,2,-3) \).

\(\mathrm{ABCD}\) is a trapezium with AB parallel to \(\mathrm{DC}\) and \(\mathrm{DC}=3 \mathrm{AB} . \mathrm{M}\) is the midpoint of \(\mathrm{DC} \cdot \overline{\mathrm{AB}}=\overline{\mathrm{p}}, \overline{\mathrm{BC}}=\overline{\mathrm{q}}\). Find \(\overline{\mathrm{MB}}\) in terms of \(\overline{\mathrm{p}}\) and \(\overline{\mathrm{q}}\):

Find the angle between two vectors \((\vec{a}=2 \hat{i}+\hat{j}-3 \hat{k})\) and \((\vec{b}=3 \hat{i}-2 \hat{j}-\hat{k})\).

If \(\vec{A}=\hat{i}+6 i+6 k, \vec{B}=-4 \hat{i}+9 \hat{i}+6 \hat{k}, \vec{G}=\frac{-5}{3} \hat{i}+\frac{22}{3} \hat{j}+\frac{22}{3} \hat{k} .\) \(G\) is the centroid then value of C is:

If the vectors \(a \vec{i}+\vec{j}+\vec{k}, \vec{i}+b \vec{j}+\vec{k}, \vec{i}+\vec{j}+c \vec{k}~(a, b, c\) are not equal to 1\()\) are coplanar, then \(\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}=? \)

The vector \(\overrightarrow{a}=\alpha \hat{i}+2 \hat{j}+\beta \hat{k}\) lies in the plane of the vectors \(\overrightarrow{b}=\hat{i}+\hat{j}\) and \(\overrightarrow{c}=\hat{i}+\hat{k}\) and bisects the angle between \(b\) and \(c\). Then, which one of the following gives possible values of \(\alpha\) and \(\beta\) is:

If \(\vec{a}, \vec{b}, \vec{c}\) are the position vectors of the points \(A, B, C\) respectively and \(2 \vec{a}+\) \(3 \vec{\mathrm{b}}-5 \vec{\mathrm{c}}={0}\), then find the ratio in which the point \(\mathrm{C}\) divides line segment AB.

Let the unit vectors a and \(b\) be perpendicular to each other and the unit vector \(c\) be inclined at an angle \(\theta\) to both a and b. If \(c=x a+y b+z(a \times b)\), then:

Let \(\vec{a}=\hat{j}-\hat{k}\) and \(\vec{c}=\hat{i}-\hat{j}-\hat{k}\). Then the vector \(\vec{b}\) satisfying \((\vec{a} \times \vec{b})+\vec{c}=0\) and \(\vec{a} \cdot \vec{b}=3\) is:

Find the position vector of the midpoint of the vector joining the points \(\mathrm{P}(2,3,4)\) and \(\mathrm{Q}(4,1,-2)\).