Vector Algebra Questions - JEE

When |\mathbf{c} - \mathbf{a}| attains least value, then the value of |\mathbf{c}| is

Let $\vec{a}=\hat{i}-3\hat{j}+7\hat{k}$, $\vec{b}=2\hat{i}-\hat{j}+\hat{k}$ and $\vec{c}$ be a vector such that $(\vec{a}+2\vec{b})\times\vec{c}=3(\vec{c}\times\vec{a})$. If $\vec{a}\cdot\vec{c}=130$, then $\vec{b}\cdot\vec{c}$ is equal to _____

If a = i + 2j − 2k, b = 2i − j + k and c = i + 3j − k, then a × (b × c) is equal to

66. Let $\vec{a} = \hat{i} + \hat{j} + \sqrt{2}\hat{k}$, $\vec{b} = b_1\hat{i} + b_2\hat{j} + \sqrt{2}\hat{k}$ and $\vec{c} = 5\hat{i} + \hat{j} + \sqrt{2}\hat{k}$ be three vectors such that projection vector of $\vec{b}$ on $\vec{a}$ is $\vec{a}$. If $\vec{a} + \vec{b}$ is perpendicular to $\vec{c}$, then $|\vec{b}|$ is equal to ________.

Let $\vec{a}=4\hat{i}+3\hat{j}$ and $\vec{b}=3\hat{i}-4\hat{j}+5\hat{k}$. If $(\vec{a}+\vec{b})\perp(\lambda\vec{a}-\vec{b})$, find $\lambda$.

If $\vec{c} \cdot \vec{d} = 0$, $5|a|^2 + 6\vec{a}\cdot\vec{b} - 8|b|^2 = 0$, find $\hat{a}\cdot\hat{b}$.

If $\vec{a}, \vec{b}, \vec{c}$ are non-coplanar vectors and $\lambda$ is a real number then $[\lambda(\vec{a}+\vec{b})\ \lambda^2\vec{b}\ \lambda\vec{c}] = [\vec{a}\ \vec{b}+\vec{c}\ \vec{b}]$ for

Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}$, $\vec{c}=\hat{j}-\hat{k}$ and a vector $\vec{b}$ be such that $\vec{a}\times\vec{b}=\vec{c}$ and $\vec{a}\cdot\vec{b}=3$. Then $|\vec{b}|$ equals

Let a = $2\mathbf{i} + \mathbf{j} + \mathbf{k}$, b = $\mathbf{i} + 2\mathbf{j} - \mathbf{k}$ and c is a unit vector coplanar to them. If c is perpendicular to a, then c is equal to

Let v = $2\mathbf{i} + \mathbf{j} - \mathbf{k}$ and w = $\mathbf{i} + 3\mathbf{k}$. If u is a unit vector and the maximum value of $[\mathbf{u}, \mathbf{v}, \mathbf{w}] = \lambda$, then the value of $\lambda - 51$ is

A unit vector a makes an angle $\frac{\pi}{4}$ with the Z-axis. If a + i + j is a unit vector, then a is equal to

Let $\vec{a} = \vec{i} + \vec{j} + \vec{k}$, $\vec{b} = \vec{i} - \vec{j} + \vec{k}$ and $\vec{c} = \vec{i} - \vec{j} - \vec{k}$ be three vectors. A vector $\vec{v}$ in the plane of $\vec{a}$ and $\vec{b}$, whose projection on $\vec{c}$ is $\frac{1}{3}$, is given by

Ex. 97 Let a = 2i + 3j - 6k, b = 2i - 3j + 6k and c = -2i + 3j + 6k. Let a₁ be the projection of a on b and a₂ be the projection of a₁ on c. Then a₂ is equal to

If $a_1$ and $a_2$ are two values of $a$ for which the unit vector $\vec{a} = a\hat{i} + 2a\hat{j} - \frac{1}{2}\hat{k}$ is linearly dependent with $\hat{i} + b\hat{j} - 2\hat{k}$, then $\frac{1}{a_1} + \frac{1}{a_2}$ is equal to

If internal and external bisectors of ∠A of △ABC meet the base BC at D and E respectively (D and E lie on same side of B), then which relation holds?

If $a^2 + b^2 + c^2 = 1$ where $a, b, c \in \mathbb{R}$, then the maximum value of $(4a - 3b)^2 + (5b - 4c)^2 + (3c - 5a)^2$ is

Vectors b = (tan α, −1, 2 sin α/2) and c = (tan α, tan α, −3 sin α/2) are orthogonal and vectors a = (1, 3, sin 2α) makes an obtuse angle with the Z-axis, then the value of α is

The vectors $u = (al + a_1l_1)\mathbf{i} + (am + a_1m_1)\mathbf{j} + (an + a_1n_1)\mathbf{k}$, $v = (bl + b_1l_1)\mathbf{i} + (bm + b_1m_1)\mathbf{j} + (bn + b_1n_1)\mathbf{k}$ and $w = (cl + c_1l_1)\mathbf{i} + (cm + c_1m_1)\mathbf{j} + (cn + c_1n_1)\mathbf{k}$

The vectors $\vec{a}$ and $\vec{b}$ are not perpendicular and $\vec{c}$ and $\vec{d}$ are two vectors satisfying: $\vec{b} \times \vec{c} = \vec{b} \times \vec{d}$ and $\vec{a} \cdot \vec{d} = 0$. Then the vector $\vec{d}$ is equal to

The value of a so that the volume of the parallelepiped formed by i + aj + k, j + ak, and ai + k becomes minimum is

Given a = 3î + 2ĵ + xk̂ and b = î − ĵ + k̂. Then the minimum value of |a × b| is

If $|\vec{c}|^2 = 60$ and $\vec{c} \times (\hat{i} + 2\hat{j} + 5\hat{k}) = \vec{0}$, then a value of $\vec{c} \cdot (-7\hat{i} + 2\hat{j} + 3\hat{k})$ is

Three forces P, Q, R act along the bisectors of the angles of a triangle ABC. By Lami's theorem, which of the following is correct?

Vector Algebra Questions (573)