Mock Test 2 — Vector Algebra

If $\mathbf{a} = \hat{i} + \hat{j} + \hat{k}$, $\mathbf{b} = \hat{i} - \hat{j} + 2\hat{k}$, $\mathbf{c} = x\hat{i} + (x-2)\hat{j} - \hat{k}$, and $[\mathbf{a}\,\mathbf{b}\,\mathbf{c}] = 0$, then $x$ equals:

Let $\mathbf{a}, \mathbf{b}, \mathbf{c}$ be unit vectors such that $\mathbf{a} \cdot \mathbf{b} = \mathbf{a} \cdot \mathbf{c} = 0$ and the angle between $\mathbf{b}$ and $\mathbf{c}$ is $\dfrac{\pi}{6}$. Then $\mathbf{a}$ equals:

For three vectors $\mathbf{u}, \mathbf{v}, \mathbf{w}$ which of the following is NOT generally equal to the others?

If $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are non-coplanar vectors and $\mathbf{a} \times \mathbf{b} = \mathbf{c}$, $\mathbf{b} \times \mathbf{c} = \mathbf{a}$, $\mathbf{c} \times \mathbf{a} = \mathbf{b}$, then $|\mathbf{a}| + |\mathbf{b}| + |\mathbf{c}|$ equals:

The volume of the tetrahedron with vertices at $(0,0,0)$, $(1,2,3)$, $(2,-1,1)$, $(3,1,-2)$ is:

If $\mathbf{a} \times \mathbf{b} = \mathbf{c} \times \mathbf{d}$ and $\mathbf{a} \times \mathbf{c} = \mathbf{b} \times \mathbf{d}$, then $\mathbf{a} - \mathbf{d}$ is parallel to:

Let $\mathbf{a} = 2\hat{i} + \hat{j} - 2\hat{k}$ and $\mathbf{b} = \hat{i} + \hat{j}$. If $\mathbf{c}$ is a vector such that $\mathbf{a} \cdot \mathbf{c} = |\mathbf{c}|$, $|\mathbf{c} - \mathbf{a}| = 2\sqrt{2}$, and the angle between $\mathbf{a} \times \mathbf{b}$ and $\mathbf{c}$ is $30°$, then $|(\mathbf{a} \times \mathbf{b}) \times \mathbf{c}|$ equals:

If $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are vectors such that $[\mathbf{a}\,\mathbf{b}\,\mathbf{c}] = 4$, then $[\mathbf{a} \times \mathbf{b}\,\,\mathbf{b} \times \mathbf{c}\,\,\mathbf{c} \times \mathbf{a}]$ equals:

Let $\mathbf{a}$ and $\mathbf{b}$ be two unit vectors. If the vectors $\mathbf{c} = \mathbf{a} + 2\mathbf{b}$ and $\mathbf{d} = 5\mathbf{a} - 4\mathbf{b}$ are perpendicular to each other, then the angle between $\mathbf{a}$ and $\mathbf{b}$ is:

If $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ are unit vectors satisfying $\mathbf{a} + \mathbf{b} + \mathbf{c} = \mathbf{0}$ and the magnitude of $\mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{c} + \mathbf{c} \times \mathbf{a}$ equals:

If $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are three vectors of magnitudes $1, 2, 3$ respectively, and they are mutually perpendicular, then $|\mathbf{a} + \mathbf{b} + \mathbf{c}|^2 = N$. Find the value of $N$.

The vectors $\mathbf{a} = \hat{i} - 2\hat{j} + 3\hat{k}$, $\mathbf{b} = -2\hat{i} + 3\hat{j} - 4\hat{k}$, $\mathbf{c} = \hat{i} - 3\hat{j} + 5\hat{k}$ are coplanar. Find the number $\lambda$ (if any) such that $\mathbf{a} - \lambda\mathbf{b} - (1+\lambda)\mathbf{c}$ becomes the zero vector, expressed as the closest integer.

In a parallelepiped with concurrent edges $\mathbf{a}, \mathbf{b}, \mathbf{c}$, if $|\mathbf{a}| = |\mathbf{b}| = |\mathbf{c}| = 1$ and all the angles between consecutive edges equal $60°$, then the square of the volume of the parallelepiped equals $\dfrac{N}{4}$. Find $N$.

Assertion (A): If $\mathbf{a} \times \mathbf{b} = \mathbf{a} \times \mathbf{c}$ and $\mathbf{a} \neq \mathbf{0}$, then $\mathbf{b} = \mathbf{c}$.
Reason (R): The cross product equation implies $\mathbf{a} \times (\mathbf{b} - \mathbf{c}) = \mathbf{0}$, which only guarantees $\mathbf{b} - \mathbf{c}$ is parallel to $\mathbf{a}$.

Assertion (A): For any three vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$, the identity $\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) + \mathbf{b} \times (\mathbf{c} \times \mathbf{a}) + \mathbf{c} \times (\mathbf{a} \times \mathbf{b}) = \mathbf{0}$ holds (Jacobi identity).
Reason (R): This follows by expanding each term using the BAC--CAB rule and observing the cancellation.