List of top Mathematics Questions on types of vectors

If $\vec{a} = 4\hat{i} + \lambda \hat{j} - 6\hat{k}$ and $\vec{b} = -6\hat{i} + 12\hat{j} + 9\hat{k}$ are collinear, then the value of $\lambda$ is equal to

If $y = 4\sqrt{x}$ then $\frac{d^2y}{dx^2} =$

Let $ \bar{a} $ be a vector perpendicular to the plane containing non-zero vectors $ \bar{b} $ and $ \bar{c} $. If $ \bar{a}, \bar{b}, \bar{c} $ are such that

\[ |\bar{a} + \bar{b} + \bar{c}| = \sqrt{|\bar{a}|^2 + |\bar{b}|^2 + |\bar{c}|^2}, \]

then

\[ |(\bar{a} \times \bar{b}) \cdot (\bar{a} \times \bar{c})| = \]

If $ A(1, 2, -3) $, $ B(2, 3, -1) $, and $ C(3, 1, 1) $ are the vertices of triangle $ \Delta ABC $, then find $ \left| \frac{\cos A}{\cos B} \right|.$

In a triangle $ABC$, if \[ r_1 r_2 + rr_3 = 35, \quad r_2 r_3 + rr_1 = 63, \quad r_3 r_1 + rr_2 = 45 \] then $2s$ is:

Let $ \bar{a}, \bar{b}, \bar{c} $ be three vectors each having $ \sqrt{2} $ magnitude such that

\[ (\bar{a}, \bar{b}) = (\bar{b}, \bar{c}) = (\bar{c}, \bar{a}) = \frac{\pi}{3}. \]

If

\[ \bar{x} = \bar{a} \times (\bar{b} \times \bar{c}) \quad \text{and} \quad \bar{y} = \bar{b} \times (\bar{c} \times \bar{a}), \]

then:

If $ a, b, c $ are the intercepts made on X, Y, Z-axes respectively by the plane passing through the points $ (1, 0, -2) $, $ (3, -1, 2) $, and $ (0, -3, 4) $, then $ 3a + 4b + 7c = $?

A vector of magnitude $\sqrt{2}$ units along the internal bisector of the angle between the vectors $2\mathbf{i} - 2\mathbf{j} + \mathbf{k}$ and $\mathbf{i} + 2\mathbf{j} + 2\mathbf{k}$ is:

A plane $ \pi_1 $ passing through the point $ 3\hat{i} - 7\hat{j} + 5\hat{k} $ is perpendicular to the vector $ \hat{i} + 2\hat{j} - 2\hat{k} $ and another plane $ \pi_2 $ passing through the point $ 2\hat{i} + 7\hat{j} - 8\hat{k} $ is perpendicular to the vector $ 3\hat{i} + 2\hat{j} + 6\hat{k} $. If $ p_1 $ and $ p_2 $ are the perpendicular distances from the origin to the planes $ \pi_1 $ and $ \pi_2 $ respectively, then $ p_1 - p_2 $ is:

In triangle ABC, if $ A = 45^\circ $, $ C = 75^\circ $, and $ R = \sqrt{2} $, then the value of $ r $ is:

Given two planes with equations $ \mathbf{r} \cdot (\mathbf{i} - \mathbf{j} + \mathbf{k}) = 5 $ and $ \mathbf{r} \cdot (2\mathbf{i} + \mathbf{j} - \mathbf{k}) = 3 $. A plane $ \pi $ passing through the line of intersection of these two planes also passes through the point (0,1,2). If the equation of $ \pi $ is $ \mathbf{r} \cdot (a\mathbf{i} + b\mathbf{j} + c\mathbf{k}) = m $, determine the value of $ \frac{bc}{a^2} $:

$ \vec{a}, \vec{b}, \vec{c} $ are three vectors such that $|\vec{a}| = 3$, $|\vec{b}| = 2\sqrt{2}$, $|\vec{c}| = 5$, and $ \vec{c} $ is perpendicular to the plane of $ \vec{a} $ and $ \vec{b} $.

If the angle between the vectors $ \vec{a} $ and $ \vec{b} $ is $ \frac{\pi}{4} $, then

\[ |\vec{a} + \vec{b} + \vec{c}| = \ ? \]

If $ \vec{a}, \vec{b}, \vec{c} $ are non-coplanar vectors and the points \[ \lambda \vec{a} + 3\vec{b} - \vec{c}, \quad \vec{a} - \lambda \vec{b} + 3\vec{c}, \quad 3\vec{a} + 4\vec{b} - \lambda \vec{c}, \quad \vec{a} - 6\vec{b} + 6\vec{c} \] are coplanar, then one of the values of $ \lambda $ is:

P and Q are the points of trisection of the line segment AB. If the position vectors of A and B are $ 2\hat{i} - 5\hat{j} + 3\hat{k} $ and $ 4\hat{i} + \hat{j} - 6\hat{k} $ respectively, then the position vector of the point that divides PQ in the ratio 2:3 is:

If $ \mathbf{a} = 2\overline{i} - \overline{j}, \overline{b} = 2\overline{j} - \overline{k}, \overline{c} = 2\overline{k} - \overline{i} $ are three vectors and $ \overline{d} $ is a unit vector perpendicular to $ \overline{c} $, If $ \overline{a}, \overline{b}, \overline{d} $ are coplanar vectors, then $ |\overline{d} \cdot \overline{b}| = $:

A plane $\pi$ passes through the points $ (5,1,2) $, $ (3,-4,6) $, and $ (7,0,-1) $. If $ p $ is the perpendicular distance from the origin to the plane $\pi$ and $[l, m, n]$ are the direction cosines of a normal to the plane $\pi$, then $[3l + 2m + 5n] =$

A unit vector $ \mathbf{e} = a\mathbf{i} + b\mathbf{j} + c\mathbf{k} $ is coplanar with the vectors $ \mathbf{i} - 3\mathbf{j} + 5\mathbf{k} $ and $ 3\mathbf{i} + \mathbf{j} - 5\mathbf{k} $ and is perpendicular to the vector $ \mathbf{i} + \mathbf{j} + \mathbf{k} $. Calculate $2a^2 + 3b^2 + 4c^2$:

Given the vectors:

\[ \mathbf{a} = \mathbf{i} + 2\mathbf{j} + \mathbf{k} \]

\[ \mathbf{b} = 3(\mathbf{i} - \mathbf{j} + \mathbf{k}) = 3\mathbf{i} - 3\mathbf{j} + 3\mathbf{k} \]

where

\[ \mathbf{a} \times \mathbf{c} = \mathbf{b} \]