List of top Mathematics Questions on types of vectors asked in TS EAMCET

In triangle ABC, if \( A = 45^\circ \), \( C = 75^\circ \), and \( R = \sqrt{2} \), then the value of \( r \) 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:
If \( A(1, 2, -3) \), \( B(2, 3, -1) \), and \( C(3, 1, 1) \) are the vertices of triangle \( \Delta ABC \), then find \( \left| \frac{\cos A}{\cos B} \right|.\)
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 \( 3a + 4b + 7c = \)?
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:
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:

\( \vec{a}, \vec{b}, \vec{c} \) 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

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

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:

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

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

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 \( \frac{bc}{a^2} \):
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 \( \mathbf{a} = 4\hat{i} + 5\hat{j} - 3\hat{k} \) and \( \mathbf{b} = 6\hat{i} - 2\hat{j} - 2\hat{k} \) are two vectors, then the magnitude of the component of \( \mathbf{b} \) parallel to \( \mathbf{a} \) 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 \( |\overline{d} \cdot \overline{b}| = \):
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:
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 \([3l + 2m + 5n] =\)

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

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

\[ \mathbf{a} \cdot \mathbf{x} = 3 \]

Find:

\[ \mathbf{a} \cdot (\mathbf{x} \times \mathbf{b} - \mathbf{c}) \]

Given the position vectors of points \(A\) and \(B\) as \( \mathbf{A} = 2\mathbf{i} - 3\mathbf{j} + \mathbf{k} \) and \( \mathbf{B} = \mathbf{i} + 2\mathbf{j} - 3\mathbf{k} \), and \(C\) divides \(AB\) in the ratio 3:2. If \( \mathbf{D} = 3\mathbf{i} - \mathbf{j} + 2\mathbf{k} \) is the position vector of point \(D\), find the unit vector in the direction of \( \mathbf{CD} \):
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 vectors \( \mathbf{a} = \mathbf{i} + \mathbf{j} - 2\mathbf{k} \), \( \mathbf{b} = \mathbf{i} - 2\mathbf{j} + \mathbf{k} \), and \( \mathbf{c} = 2\mathbf{i} + \mathbf{j} - \mathbf{k} \). If \( \mathbf{d} \) is a normal to the plane of \( \mathbf{a} \) and \( \mathbf{b} \) and satisfies \( \mathbf{d} \cdot \mathbf{c} = 2 \), find the magnitude of \( \mathbf{d} \):
The foot of the perpendicular drawn from the point \((-2, -1, 3)\) to a plane is \((1, 0, -2)\). If \(a, b, c\) are the intercepts made by the plane \(\pi\) on \(X, Y, Z\)-axes respectively, then \(3a+b+5c =\)
If \(\theta\) is the angle between the vectors \(4\mathbf{i} - \mathbf{j} + 2\mathbf{k}\) and \( \mathbf{i} + 3\mathbf{j} - 2\mathbf{k}\), then \(\sin 2\theta\) is:
Given the position vectors of points \(A\) and \(B\) as \( \mathbf{A} = 2\mathbf{i} - 3\mathbf{j} + \mathbf{k} \) and \( \mathbf{B} = \mathbf{i} + 2\mathbf{j} - 3\mathbf{k} \), and \(C\) divides \(AB\) in the ratio 3:2. If \( \mathbf{D} = 3\mathbf{i} - \mathbf{j} + 2\mathbf{k} \) is the position vector of point \(D\), find the unit vector in the direction of \( \mathbf{CD} \):
The position vector of the point of intersection of the line joining the points \( \mathbf{i} - \mathbf{j} + \mathbf{k} \) and the line joining the points \( 2\mathbf{i} + \mathbf{j} - 6\mathbf{k} \), \( 3\mathbf{i} - \mathbf{j} - 7\mathbf{k} \) is:
In a triangle ABC, if \( r_1 = 6 \), \( r_2 = 9 \), \( r_3 = 18 \), then \( \cos A \) is:
The position vectors of the vertices \( A, B, C \) of a triangle are given as: \[ \vec{A} = \hat{i} - 2\hat{j} + k, \quad \vec{B} = 2\hat{i} + \hat{j} - k, \quad \vec{C} = \hat{i} - \hat{j} - 2k \] If \( D \) and \( E \) are the midpoints of \( BC \) and \( CA \) respectively, then the unit vector along \( \overrightarrow{DE} \) is: