List of top Mathematics Questions on Unit Vectors

Let \( \mathbf{a} \) and \( \mathbf{b} \) be two unit vectors such that the angle between them is \( \frac{\pi}{3} \). If \( \lambda \mathbf{a} + 2 \mathbf{b} \) and \( 3 \mathbf{a} - \lambda \mathbf{b} \) are perpendicular to each other, then the number of values of \( \lambda \) in \( [-1, 3] \) is:
\( \vec{a}, \vec{b}, \vec{c} \) are three mutually perpendicular unit vectors. If \( \theta \) is the angle between \( \vec{a} \) and \( (2\vec{a} + 3\vec{b} + 6\vec{c}) \), find the value of \( \cos \theta \).
If \( |\vec{a}| = 2 \) and \( -3 \leq k \leq 2 \), then \( |k\vec{a} | \in: \)
(d) If vectors \( \mathbf{5i - \lambda j + 2k} \) and \( \mathbf{2i + 3j + 4k} \) are perpendicular, find \( \lambda \):
Solve the differential equation \[ \frac{dy}{dx} = e^x \sin x. \]
Solve the differential equation \[ \frac{dy}{dx} = \frac{1 + x^2}{1 + y^2} \]
Find the perpendicular unit vectors on the vectors \[ \mathbf{a} = 2\hat{i} - \hat{j} + \hat{k} \quad \text{and} \quad \mathbf{b} = 3\hat{i} + 4\hat{j} - \hat{k} \] and find the sine of the angle between them.
(e) The angle between the vectors \( 3\hat{i} - 2\hat{j} + \hat{k} \) and \( 2\hat{i} + \hat{j} + 3\hat{k} \) will be:
(c) If the unit vectors \( \vec{a}, \vec{b}, \vec{c} \) are such that \( \vec{a} + \vec{b} + \vec{c} = \vec{0} \), find the value of \( \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} \):

Show that the vectors \( 2\hat{i} - \hat{j} + \hat{k}, \hat{i} - 3\hat{j} - 5\hat{k}, 3\hat{i} - 4\hat{j} - 4\hat{k} \) form the vertices of a right-angled triangle. 

Find the angle between the lines \[ \frac{x+1}{-1} = \frac{y-2}{2} = \frac{z-5}{-5} \quad \text{and} \quad \frac{x+3}{-3} = \frac{y-1}{2} = \frac{z-5}{5}. \] 

If the coordinates of the points \( A \), \( B \), \( C \), and \( D \) are \( (1, 2, 3) \), \( (4, 5, 7) \), \( (-4, 3, -6) \), and \( (2, 9, 2) \) respectively, then find the angle between the lines AB and CD.
If \( |\vec{a}| = \sqrt{3} \), \( |\vec{b}| = 2 \), and \( \vec{a} \cdot \vec{b} = \sqrt{6} \), then the angle between the vectors \( \vec{a} \) and \( \vec{b} \) is:
If the position vectors of the points \( A, B, C, D \) are successively \( \hat{i} + \hat{j} + \hat{k} \), \( 2\hat{i} + 5\hat{j} \), \( 3\hat{i} + 2\hat{j} - 3\hat{k} \) and \( \hat{i} - 6\hat{j} + \hat{k} \), then find the angle between the lines \( AB \) and \( CD \).
Find the area of a parallelogram whose adjacent sides are \( \vec{a} = 3\hat{i} + \hat{j} + 4\hat{k} \) and \( \vec{b} = \hat{i} - \hat{j} + \hat{k} \).
The angle between two vectors \( \vec{a} \) and \( \vec{b} \) is 0 and \( |\vec{a} \cdot \vec{b}| = |\vec{a} \times \vec{b}| \) is given. Find the value of \( \theta \).
Prove that \[ \int_0^{\pi/2} \log \sin x \, dx = -\frac{\pi}{2} \log 2. \]
(a) Find the direction cosines of the vector \( \hat{i} + \hat{j} - 2\hat{k} \).
(c) If the vectors \( \vec{v_1} = 3\hat{i} + 2\hat{j} + \hat{k} \) and \( \vec{v_2} = \hat{i} - 4\hat{j} + \lambda \hat{k} \) are perpendicular, find the value of \( \lambda \):
(d) The value of the expression: \[ \hat{i} \cdot \hat{i} + \hat{j} \cdot \hat{j} + \hat{k} \cdot \hat{k} \]
Solve the differential equation: \[ \frac{dy}{dx} + \frac{y}{x} = x^2, \quad y = 1 \text{ when } x = 1 \]
Solve the differential equation: \[ (\tan^{-1} y - x) \, dy = (1 + y^2) \, dx \]
(e) The degree of the differential equation \( d^2y/dx^2 = \sqrt{x} + \left(\frac{dy}{dx}\right)^2 \) is:
For which value of \( \lambda \) are the vectors \( \lambda \hat{i} + \hat{j} + \hat{k} \) and \( \hat{i} - 4\hat{j} + 2\hat{k} \) perpendicular?