Vector Algebra

Let $\vec{a} = 2\hat{i} - 3\hat{j} + \hat{k}$, $\vec{b} = 3\hat{i} + 2\hat{j} + 5\hat{k}$ and a vector $\vec{c}$ be such that $(\vec{a} - \vec{c}) \times \vec{b} = -18\hat{i} - 3\hat{j} + 12\hat{k}$ and $\vec{a} \cdot \vec{c} = 3$. If $\vec{b} \times \vec{c} = \vec{d}$, then $|\vec{a} \cdot \vec{d}|$ is equal to:

If $\vec{a}$ is a nonzero vector such that its projections on the vectors $2\hat{i}-\hat{j}+2\hat{k}$, $\hat{i}+2\hat{j}-2\hat{k}$, and $\hat{k}$ are equal, then a unit vector along $\vec{a}$ is:

Let ABCD be a tetrahedron such that the edges AB, AC and AD are mutually perpendicular. Let the areas of the triangles ABC, ACD and ADB be 5, 6 and 7 square units respectively. Then the area (in square units) of ABCD is equal to:

Consider two vectors \mathbf{u}=3\mathbf{i}-\mathbf{j} and \mathbf{v}=2\mathbf{i}+\mathbf{j}-\lambda\mathbf{k}, \lambda > 0. The angle between them is given by \cos⁻¹\bigl(\tfrac{\sqrt{5}}{2\sqrt{7}}\bigr). Let \mathbf{v}=\mathbf{v}_1+\mathbf{v}_2, where \mathbf{v}_1 is parallel to \mathbf{u} and \mathbf{v}_2 is perpendicular to \mathbf{u}. Then the value of |\mathbf{v}_1|² + |\mathbf{v}_2|² is:

Let the vectors $a$ and $b$ be of the same magnitude such that $\frac{\lvert a + b\rvert + \lvert a - b\rvert}{\lvert a + b\rvert - \lvert a - b\rvert} = \sqrt{2} + 1$. Then $\frac{\lvert a + b\rvert^2}{\lvert a\rvert^2}$ is:

Let the angle $\theta$, $0 < \theta < \frac{\pi}{2}$, be the angle between two unit vectors $\hat{a}$ and $\hat{b}$, and $\theta = \sin^{-1}\left(\frac{\sqrt{65}}{9}\right)$. If the vector $\vec{c} = 3\hat{a} + 6\hat{b} + 9(\hat{a} \times \hat{b})$, then the value of $9(\vec{c} \cdot \hat{a}) - 3(\vec{c} \cdot \hat{b})$ is:

Let $\vec{a} = \hat{i} + 2\hat{j} + \hat{k}$ and $\vec{b} = 2\hat{i} + \hat{j} - \hat{k}$. Let $\vec{c}$ be a unit vector in the plane of the vectors $\vec{a}$ and $\vec{b}$ and be perpendicular to $\vec{a}$. Then such a vector $\vec{c}$ is:

Let the arc $AC$ of a circle subtend a right angle at the centre $O$. If the point $B$ on the arc $AC$, divides the arc $AC$ such that $\frac{\text{length of arc } AB}{\text{length of arc } BC} = \frac{1}{5}$, and $\overrightarrow{OC} = \alpha \overrightarrow{OA} + \beta \overrightarrow{OB}$, then $\alpha + \sqrt{2}(\sqrt{3} - 1) \beta$ is equal to

Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that the angle between them is $\frac{\pi}{3}$. If $\lambda \vec{a} + 2 \vec{b}$ and $3 \vec{a} - \lambda \vec{b}$ are perpendicular to each other, then the number of values of $\lambda$ in $[-1,3]$ is:

Let $\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$, $\vec{b} = 3\hat{i} + \hat{j} - \hat{k}$ and $\vec{c}$ be three vectors such that $\vec{c}$ is coplanar with $\vec{a}$ and $\vec{b}$. If the vector $\vec{c}$ is perpendicular to $\vec{b}$ and $\vec{a} \cdot \vec{c} = 5$, then $|\vec{c}|$ is equal to