Vector Algebra - Algebra - Mathematics Previous Year Questions

If \(|\mathbf{a}|=16,|\mathbf{b}|=4\), then \(\sqrt{|\mathbf{a} \times \mathbf{b}|^2+|\mathbf{a} \cdot \mathbf{b}|^2}=\)

A unit vector perpendicular to the plane containing the vector \(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(-2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}}\) is

If the angle between \(\mathbf{a}\) & \(\mathbf{b}\) is \(\frac{2 \pi}{3}\) and the projection of \(\mathbf{a}\) in the direction of \(\mathbf{b}\) is \(-\)2 , the \(|\mathbf{a}|=\)

$$[\mathbf{a}+2 \mathbf{b}-\mathbf{c}, \mathbf{a}-\mathbf{b}, \mathbf{a}-\mathbf{b}-\mathbf{c}]=$$

The two vector \(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(\hat{\mathbf{i}}+3 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}\) represent the two sides \(\overline{A B}\) and \(\overline{A C}\) respectively of a \(\triangle A B C\). The length of the median through \(A\) is

If \(|\mathbf{a}+\mathbf{b}|^2+|\mathbf{a} \cdot \mathbf{b}|^2=144|\mathbf{a}|=6\), then \(|\mathbf{b}|\) is equal to

If \(\mathbf{a}\) and \(\mathbf{b}\) are unit vectors and \(\theta\) is the angle between \(\mathbf{a}\) and \(\mathbf{b}\), then \(\sin \frac{\theta}{2}\) is equal to

If the vectors \(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}, 2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}\) and \(\lambda \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}\) are coplanar, then the value of \(\lambda\) is

A vector a makes equal acute angles on the coordinate axis. Then the projection of vector \(\mathbf{b}=5 \hat{\mathbf{i}}+7 \hat{\mathbf{j}}+\hat{\mathbf{k}}\) on \(\mathbf{a}\) is

If \(\mathbf{a} \cdot \mathbf{b}=0\) and \(\mathbf{a}+\mathbf{b}\) makes an angle \(60^{\circ}\) with \(a\), then

The diagonals of a parallelogram are the vectors \(3 \hat{\mathbf{i}}+6 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}\). and \(-\hat{\mathbf{i}}-2 \hat{\mathbf{j}}-8 \hat{\mathbf{k}}\). Then the length of the shorter side of parallelogram is

If the area of the parallelogram with \(\mathbf{a}\) and \(\mathbf{b}\) as two adjacent sides is 15 sq units, then the area of the parallelogram having \(\mathrm{3 a+2 b}\) and \(\mathbf{a}+3 \mathbf{b}\) as two adjacent sides in sq units is

If \(|\mathbf{a}|=2\) and \(|\mathbf{b}|=3\) and the angle between \(\mathbf{a}\) and \(\mathbf{b}\) is \(120^{\circ}\), then the length of the vector \(\left|\frac{\mathbf{a}}{2}-\frac{\mathbf{b}}{3}\right|\) is