Vector Algebra Questions (573)

Let \(\alpha,\beta,\gamma\) be distinct real numbers. The points whose position vectors are \(\vec{a}=\alpha\hat{i}+\beta\hat{j}+\gamma\hat{k}\), \(\vec{b}=\beta\hat{i}+\gamma\hat{j}+\alpha\hat{k}\), \(\vec{c}=\gamma\hat{i}+\alpha\hat{j}+\beta\hat{k}\)

[JEE Main 2021] Suppose \(\vec{a},\vec{b},\vec{c}\) are unit vectors and \((\vec{a}+3\vec{b})\perp\vec{c}\) and \((\vec{a}+\vec{b})\perp(\vec{a}+3\vec{b})\). Then which of the following is true?

[JEE Main 2021] Let \(\vec{a},\vec{b},\vec{c}\) be three mutually perpendicular unit vectors. The angle \(\theta\) between each of them and the vector \(\vec{a}+\vec{b}+\vec{c}\) is

Let \(\vec{a}=q_1\hat{i}+q_2\hat{j}+q_3\hat{k}\) make equal angles with OX, OY, OZ and \(|\vec{a}|=\sqrt{3}\). If the projection of \(\vec{a}\) on \(\hat{i}+\hat{j}-\hat{k}\) is 1, find \(q_1+q_2+q_3\).

\(A,B,C,D\) are four points in a plane with position vectors \(\vec{a},\vec{b},\vec{c},\vec{d}\) respectively such that \((\vec{a}-\vec{d})\cdot(\vec{b}-\vec{c})=0\) and \((\vec{b}-\vec{d})\cdot(\vec{c}-\vec{a})=0\). Then \(D\) is the

[JEE Main 2022] Let \(\vec{a}=\hat{i}+\hat{j}+\hat{k}\) and \(\vec{b}=2\hat{i}+\hat{j}-\hat{k}\). If \(\vec{p}\) is such that \(\vec{a}\times\vec{p}=\vec{b}\) and \(\vec{a}\cdot\vec{p}=0\), then the value of \([\vec{a},\vec{p},\vec{b}]\) is

For \(p>0\), the vector \(\vec{v}_2=(2,\,-(\sqrt{3}\,p+1))\) is obtained by rotating \(\vec{v}_1=\sqrt{3}(-1,\,-(p^2+\sqrt{3}))\) about the origin counter-clockwise. If the angle of rotation is \(\theta\), find \(\tan\theta\).

Let a vector \(\hat{i}+\sqrt{2}\,\hat{j}+\sqrt{2}\,\hat{k}\) be obtained by rotating the vector \(\sqrt{3}\,\hat{j}\) by an angle \(45°\) about the origin in the clockwise direction to the first quadrant. Then the area of the triangle formed by the vector \((\hat{i}+\sqrt{2}\,\hat{j}+\sqrt{2}\,\hat{k})\) with the coordinate axes is equal to:

Consider points \(A,B,C\) with position vectors \(\vec{a},\vec{b},\vec{c}\) respectively. Statement-1: \(\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CA}=\vec{0}\). Statement-2: \(A,B,C\) form the vertices of a triangle.

Vectors \(\vec{a}\) and \(\vec{b}\) make an angle \(\theta=\frac{2\pi}{3}\). If \(|\vec{a}|=1,\;|\vec{b}|=2\), then the minimum value of \(|2\vec{a}+\vec{b}|^2+|2\vec{a}-\vec{b}|^2\) over all configurations is

The altitude of a parallelepiped whose three coterminous edges are \(\vec{A}=\hat{i}+\hat{j}+\hat{k}\), \(\vec{B}=2\hat{i}+4\hat{j}-\hat{k}\), \(\vec{C}=\hat{i}+\hat{j}+3\hat{k}\), with \(\vec{A}\) and \(\vec{B}\) as the base, is

If \(\hat{a}\) and \(\hat{b}\) are unit vectors and \(|\hat{a}+\hat{b}|=\sqrt{3}\), find \((2\hat{a}-\hat{b})\cdot(3\hat{a}+2\hat{b})\).

Let \(\vec{a},\vec{b}\) be two non-zero non-collinear vectors. Then for any scalar \(\lambda\neq0\), \(\vec{a}=\lambda\vec{b}\) if and only if

A line passes through point \(A\) with position vector \(\vec{v}=\hat{j}-\hat{k}\) and is parallel to the vector \(\hat{i}+\hat{j}\). For any point \(P\) on this line, which of the following is/are true?

[JEE Main 2019] Let \(\vec{a}=\hat{i}-\hat{j}\) and \(\vec{b}=-\hat{i}+\hat{j}+\hat{k}\) be two given vectors. Let \(\vec{c}=\vec{a}\times\vec{b}\). Then which of the following is NOT true?

The altitude of a parallelepiped whose three coterminous edges are \(\vec{A}=\hat{i}+\hat{j}+\hat{k}\), \(\vec{B}=2\hat{i}+4\hat{j}-\hat{k}\), \(\vec{C}=\hat{i}+\hat{j}+3\hat{k}\), with \(\vec{A}\) and \(\vec{B}\) as the base, is

[JEE Main 2021] If a unit vector \(\hat{a}\) makes angles \(\dfrac\pi3\) with \(\hat{i}\), \(\dfrac\pi4\) with \(\hat{j}\) and an acute angle \(\theta\) with \(\hat{k}\), then \(\theta\) equals

\(\vec{a}, \vec{b}, \vec{c}\) are 3 vectors, such that \(\vec{a} + \vec{b} + \vec{c} = \vec{0}\), \(|\vec{a}| = 1\), \(|\vec{b}| = 2\), \(|\vec{c}| = 3\), then \(\vec{a}\cdot\vec{b} + \vec{b}\cdot\vec{c} + \vec{c}\cdot\vec{a}\) is equal to

A velocity 1/4 m/s is resolved into two components along \(OA\) and \(OB\) making angles 30° and 45°, respectively, with the given velocity. Then the component along \(OB\) is

The resultant \(R\) of two forces acting on a particle is at right angles to one of them and its magnitude is one third of the other force. The ratio of larger force to smaller one is

[JEE Main 2019] Let \(\vec{a}\) and \(\vec{b}\) be unit vectors and \(\alpha\) be the angle between them. Then \(\vec{a}+\vec{b}\) is a unit vector if

[JEE Main 2022] Let \(\vec{a}=\hat{i}+2\hat{j}+3\hat{k}\), \(\vec{b}=\hat{i}-\hat{j}+2\hat{k}\), \(\vec{c}=5\hat{i}+3\hat{j}-\hat{k}\). If \(\alpha\) is the projection of \((\vec{a}+\vec{b})\) on \(\vec{c}\), and \(\beta\) is the projection of \(\vec{c}\) on \((\vec{a}+\vec{b})\), find \(6(\alpha+\beta)\).

The non-zero vectors \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\) are related by \(\vec{a} = 8\vec{b}\) and \(\vec{c} = -7\vec{b}\). Then the angle between \(\vec{a}\) and \(\vec{c}\) is