91Ó°ÊÓ

If position vector of points \(A, B\) and \(C\) are respectively \(\hat{i}, \hat{j}\) and \(\hat{\mathbf{k}}\) and \(A B=C X\), then position vector of point \(X\) is \((a)-\hat{i}+\hat{j}+\hat{k}\) (b) \(\hat{i}-\hat{j}+\hat{k}\) (c) \(\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}\) (d) \(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}\)

The position vectors of \(A\) and \(B\) are \(2 \hat{\mathbf{i}}-9 \hat{\mathbf{j}}-4 \hat{\mathbf{k}}\) and \(6 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}-8 \hat{\mathbf{k}}\) respectively, then the magnitude of \(\mathbf{A B}\) is (a) 11 (b) 12 (c) 13 (d) 14

If the position vectors of \(P\) and \(Q \operatorname{are}(\hat{\mathbf{i}}+3 \hat{\mathbf{j}}-7 \hat{\mathbf{k}})\) and \((5 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+4 \hat{\mathbf{k}})\), then \(|\mathbf{P Q}|\) is (a) \(\sqrt{158}\) (b) \(\sqrt{160}\) (c) \(\sqrt{161}\) (d) \(\sqrt{162}\)

If the position vectors of \(P\) and \(Q\) are \(\hat{i}+2 \hat{\mathbf{j}}-7 \hat{\mathbf{k}}\) and \(5 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}\) respectively, the cosine of the angle between \(\mathrm{PQ}\) and \(Z\)-axis is (a) \(\frac{4}{\sqrt{162}}\) (b) \(\frac{11}{\sqrt{162}}\) (c) \(\frac{5}{\sqrt{162}}\) (d) \(\frac{-5}{\sqrt{162}}\)

If the position vectors of \(A\) and \(B\) are \(\hat{\mathbf{i}}+3 \hat{j}-7 \hat{\mathbf{k}}\) and \(5 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}\), then the direction cosine of \(\mathbf{A B}\) along \(Y\)-axis is (a) \(\frac{4}{\sqrt{162}}\) (b) \(-\frac{5}{\sqrt{162}}\) (c) \(-5\) (d) 11

The direction cosines of vector \(\mathbf{a}=3 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}\) in the direction of positive axis of \(X\), is (a) \(\pm \frac{3}{\sqrt{50}}\) (b) \(\frac{4}{\sqrt{50}}\) (c) \(\frac{3}{\sqrt{50}}\) (d) \(-\frac{4}{\sqrt{50}}\)

The direction cosines of the vector \(3 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}\) are (a) \(\frac{3}{5},-\frac{4}{5}, \frac{1}{5}\) (b) \(\frac{3}{5 \sqrt{2}} \cdot \frac{-4}{5 \sqrt{2}}, \frac{1}{\sqrt{2}}\) (c) \(\frac{3}{\sqrt{2}}, \frac{-4}{\sqrt{2}}, \frac{1}{\sqrt{2}}\) (d) \(\frac{3}{5 \sqrt{2}} \cdot \frac{4}{5 \sqrt{2}}, \frac{1}{\sqrt{2}}\)

The point having position vectors \(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}\), \(3 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}\) and \(4 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}\) are the vertices of (a) right angled triangle (b) isosceles triangle (c) equilateral triangle (d) collinear

If the position vectors of the vertices \(A, B\) and \(C\) of a \(\Delta A B C\) are \(7 \hat{\mathbf{j}}+10 \mathbf{k},-\hat{\mathbf{i}}+6 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}\) and \(-4 \hat{\mathbf{i}}+9 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}\) respectively. The triangle is (a) equilateral (b) isosceles (c) scalene (d) right angled and isosceles also

If \(\mathbf{a}, \mathbf{b}\) and \(\mathbf{c}\) are the position vectors of the vertices \(A, B\) and \(C\) of the \(\triangle A B C\), then the centroid of \(\triangle A B C\) is (a) \(\frac{a+b+c}{3}\) (b) \(\frac{1}{2}\left(\mathrm{a}+\frac{\mathbf{b}+\mathrm{c}}{2}\right)\) (c) \(a+\frac{b+c}{2}\) (d) \(\frac{a+b+c}{2}\)