Vector Algebra Questions (573)

Let \(\vec{a},\vec{b}\) be two non-zero non-collinear vectors. Then \(|\vec{a}\times\vec{b}|\) equals

Let \to ^ ^ ^ a = 3 i - j + 2k, \to ^ ^ b = a \times ( i - 2k) and \to ^ c = b \times k . Then the projection of \to ^ c - 2j \to is : on a

Let a = 2 ^i - ^j + 3 k^ , b = 3 ^i - 5 ^j + k^ and c be a vector such that a \times c = c \times b and ​ ​ ( a + c ) ⋅ ( b + c ) = 168. Then the maximum value of ∣ c ∣ 2 is :

The point having position vectors 2i + 3j + 4k, 3i + 4j + 2k and 4i + 2j + 3k are the vertices of

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{b}|=1$ and $|\vec{b}\times\vec{a}|=2$. Then $|(\vec{b}\times\vec{a})-\vec{b}|^2$ is equal to:

Let $\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$ and $\vec{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$ be two vectors such that $|\vec{a}|=1$, $\vec{a}\cdot\vec{b}=2$ and $|\vec{b}|=4$. If $\vec{c}=2(\vec{a}\times\vec{b})-3\vec{b}$, then the angle between $\vec{b}$ and $\vec{c}$ is equal to:

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{a}|=1$, $|\vec{b}|=4$ and $\vec{a}\cdot\vec{b}=2$. If $\vec{c}=(2\vec{a}\times\vec{b})-3\vec{b}$ and the angle between $\vec{b}$ and $\vec{c}$ is $\alpha$, then $192\sin^2\alpha$ is equal to _____.

Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}$, $\vec{b}=2\hat{i}+4\hat{j}-5\hat{k}$ and $\vec{c}=x\hat{i}+2\hat{j}+3\hat{k}$, $x\in\mathbb{R}$. If $\vec{d}$ is the unit vector in the direction of $\vec{b}+\vec{c}$ such that $\vec{a}\cdot\vec{d}=1$, then $(\vec{a}\times\vec{b})\cdot\vec{c}$ is equal to

Let $ABC$ be a triangle of area $15\sqrt{2}$ and the vectors $\overrightarrow{AB}=\hat{i}+2\hat{j}-7\hat{k}$, $\overrightarrow{BC}=a\hat{i}+b\hat{j}+c\hat{k}$ and $\overrightarrow{AC}=6\hat{i}+d\hat{j}-2\hat{k}$, $d>0$. Then the square of the length of the largest side of the triangle $ABC$ is _____

If $A(3,1,-1)$, $B\left(\dfrac{5}{3},\dfrac{7}{3},\dfrac{1}{3}\right)$, $C(2,2,1)$ and $D\left(\dfrac{10}{3},\dfrac{2}{3},-\dfrac{1}{3}\right)$ are the vertices of a quadrilateral $ABCD$, then its area is

The set of all $\alpha$, for which the vectors $\vec{a}=\alpha t\hat{i}+6\hat{j}-3\hat{k}$ and $\vec{b}=t\hat{i}-2\hat{j}-2\alpha t\hat{k}$ are inclined at an obtuse angle for all $t\in\mathbb{R}$, is

In an isosceles triangle \(ABC\) with \(AB=AC\), M is the midpoint of \(BC\). Given an equilateral triangle with side \(\vec{a}\), the scalar triple product \([\vec{a},\vec{b},\vec{c}]\) (where \(\vec{a},\vec{b},\vec{c}\) are sides) equals

Let \(f : R \to (0, 1)\) be a continuous function, then which of the following pair of vectors are linearly dependent for some \(x \in (0, 1)\)?

[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?

[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

Let \(\hat{a}_1,\hat{a}_2,\ldots,\hat{a}_n\) be unit vectors in the plane such that \(\displaystyle\sum_{i=1}^{n}\hat{a}_i=\vec{0}\). Let \(k=\displaystyle\sum_{i

For any vector \(\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}\) with \(10|\vec{a}|\geq1\), consider: (A) \(|\vec{a}\times\hat{i}|^2+|\vec{a}\times\hat{j}|^2+|\vec{a}\times\hat{k}|^2=2|\vec{a}|^2\). (B) \(|\vec{a}\cdot\hat{i}|^2+|\vec{a}\cdot\hat{j}|^2+|\vec{a}\cdot\hat{k}|^2=|\vec{a}|^2\). Which is true?

[JEE Main 2020] The volume of a parallelepiped whose coterminous edges are \(\vec{u}=\hat{i}+\hat{j}+\lambda\hat{k}\), \(\vec{v}=\hat{i}+\hat{j}+3\hat{k}\), \(\vec{w}=2\hat{i}+\hat{j}+\hat{k}\) is 1 cubic unit. If \(\theta\) is the angle between the edges \(\vec{u}\) and \(\vec{w}\), then \(\cos\theta\) can be

Given a parallelogram \(OACB\) with \(\overrightarrow{OA}=\vec{a}\) and \(\overrightarrow{OB}=\vec{b}\). The length of the diagonal \(\overrightarrow{OC}\) is \(a\), \(b\), \(\sqrt{a^2+b^2-ab}\) or \(\sqrt{a^2+b^2+ab}\). If angle between \(\vec{a}\) and \(\vec{b}\) is \(\frac{2\pi}{3}\), then \(|\vec{a}+\vec{b}|\) is

If \([\vec{a} \times \vec{b} \quad \vec{b} \times \vec{c} \quad \vec{c} \times \vec{a}] = \lambda[\vec{a} \; \vec{b} \; \vec{c}]^2\), then \(\lambda\) is equal to

It is given that \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\) are linearly dependent, where \(\vec{a} = \hat{i}+\hat{j}+\hat{k}\), \(\vec{b} = 4\hat{i}+3\hat{j}+4\hat{k}\), \(\vec{c} = \hat{i}+\alpha\hat{j}+\beta\hat{k}\). Find \(\beta\).

If \([\vec{a}+2\vec{b}+3\vec{c}\quad \vec{b}+2\vec{c}+3\vec{a}\quad \vec{c}+2\vec{a}+3\vec{b}] = 36\), where \(\vec{a},\vec{b}\) and \(\vec{c}\) are three vectors, then the value of \(\begin{vmatrix} \vec{a}\cdot\vec{a} & \vec{a}\cdot\vec{b} & \vec{a}\cdot\vec{c} \\ \vec{a}\cdot\vec{b} & \vec{b}\cdot\vec{b} & \vec{b}\cdot\vec{c} \\ \vec{a}\cdot\vec{c} & \vec{c}\cdot\vec{b} & \vec{c}\cdot\vec{c} \end{vmatrix}\) is ________.

If \(α\) and \(β\) are two mutually perpendicular unit vectors and \(\{rα + rβ + s(α \times β)\}\), \([α + (α \times β)]\) and \(\{sα + sβ + t(α \times β)\}\) are coplanar, then \(s\) is equal to

Given $\vec{i} + \vec{j} + \vec{k}$. Two given lines intersected, if $7\vec{i} + 10\vec{j} + 13\vec{k} + s(2\vec{i} + 3\vec{j} + 4\vec{k}) = 3\vec{i} + 5\vec{j} + 7\vec{k} + t(\vec{i} + 2\vec{j} + 3\vec{k})$

Given $|\vec{a}| = 1$, $|\vec{b}| = 2$, $|\vec{c}| = 3$ and $\vec{a} \cdot \vec{b} = 0$, $\vec{b} \cdot \vec{c} = \vec{c} \cdot \vec{a}$ (as the three vectors are mutually perpendicular)

Ex. 52 In parallelogram ABCD, L is a point on BC which divides BC in the ratio 1 : 2. AL intersects BD at P. Point P divides AL in the ratio

Ex. 110: If AP, BQ and CR are the altitudes of acute \triangle ABC and 9\overrightarrow{AP} + 4\overrightarrow{BQ} + 7\overrightarrow{CR} = \mathbf{0}, then \angle ABC is equal to

Ex. 55 Let A, B, C, D, E represent vertices of a regular pentagon ABCDE with position vectors \(\vec{a}, \vec{a} + \vec{b}, \vec{b}, \lambda \vec{a}, \lambda \vec{b}\) respectively. The ratio \(\frac{AD}{BC}\) is equal to

Ex. 45 Statement I: If \(|\mathbf{a}| = 3\), \(|\mathbf{b}| = 4\) and \(|\mathbf{a} + \mathbf{b}| = 5\), then \(|\mathbf{a} - \mathbf{b}| = 5\).Statement II: The length of the diagonals of a rectangle is the same.

68. Let \(\sqrt{3}\hat{i} + \hat{j}\), \(\hat{i} + \sqrt{3}\hat{j}\) and \(\beta\hat{i} + (1-\beta)\hat{j}\) respectively be the position vectors of the points A, B and C with respect to the origin O. If the distance of C from the bisector of the acute angle between OA and OB is \(3/\sqrt{2}\), then the sum of all possible values of \(\beta\) is ________.

Let \(\vec{a}=\hat{i}+2\hat{j}+3\hat{k}\). A vector \(\vec{b}\) satisfies \(\vec{a}\cdot\vec{b}=|\vec{b}|^2\) and \(|\vec{a}-\vec{b}|^2=7\). Find \(|\vec{b}\times\vec{a}|^2\).

The value(s) of \(x\) for which the angle between vectors \(\vec{a} = (1, x^2, 9)\) and \(\vec{b} = (4, 4x-2, 2)\) is such that \(\cos\theta = \dfrac{4(1)+(4x-2)(x)+6}{\sqrt{1+x^2+9}\cdot\sqrt{16+(4x-2)^2+4}}\) and \(\sqrt{(4)^2+(4x-2)^2+4} = 2\sqrt{1+x^2+9}\), find the possible values of \(x\).

Work done = \((\vec{F_1} + \vec{F_2}) \cdot \overrightarrow{AB}\) where \(\overrightarrow{OA} = \hat{i} + 2\hat{j} + 3\hat{k}\), \(\overrightarrow{OB} = 5\hat{i} + 4\hat{j} + \hat{k}\) and \(\vec{F_1} + \vec{F_2} = 7\hat{i} + 2\hat{j} - 4\hat{k}\). Find the work done.

Let r be perpendicular to a + b + c, where r = l[a b c]a + m[c a b]b + n[a b c]c and [a b c] = 2. Find the value of l + m + n.

The points A(2-x, 2, 2), B(2, 2-y, 2), C(2, 2, 2-z) and D(1, 1, 1) are coplanar. Find the locus of P(x, y, z).

The projections of a vector on coordinate axes are \(x_2 - x_1,\ y_2 - y_1,\ z_2 - z_1\). Also, \(x_2 - x_1 = 6,\ y_2 - y_1 = -3,\ z_2 - z_1 = 2\). Find the direction cosines of the vector.

Let \(\vec{a} = \hat{i} - \hat{k}\), \(\vec{b} = x\hat{i} + \hat{j} + (1-x)\hat{k}\) and \(\vec{c} = y\hat{i} + x\hat{j} + (1+x-y)\hat{k}\). Then \([\vec{a}, \vec{b}, \vec{c}]\) depends on

It is given that \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\) are three unit vectors such that \[\vec{a} \times (\vec{b} \times \vec{c}) = \frac{\sqrt{3}}{2}(\vec{b}+\vec{c})\] Using this condition, find the angle between \(\vec{a}\) and \(\vec{b}\) (or relevant quantity as per full question).

The unit vector perpendicular to the vectors a = 6i + 2j + 3k and b = 3i - 6j - 2k is

The values of \(a\), for which the points \(A\), \(B\), \(C\) with position vectors \(2\hat{i} - \hat{j} + \hat{k}\), \(\hat{i} - 3\hat{j} - 5\hat{k}\) and \(a\hat{i} - 3\hat{j} + \hat{k}\), respectively, are the vertices of a right-angled triangle with \(C = \pi/2\) are

In a parallelogram ABCD, \(|\overrightarrow{AB}| = a\), \(|\overrightarrow{AD}| = b\) and \(|\overrightarrow{AC}| = c\), then \(\overrightarrow{DB} \cdot \overrightarrow{AB}\) has the value

We have \(|\hat{x}+\hat{y}|^2 + |\hat{y}+\hat{z}|^2 + |\hat{z}+\hat{x}|^2\) where \(\hat{x}, \hat{y}, \hat{z}\) are unit vectors making angles \(\alpha, \beta, \gamma\) with each other. Find the minimum value of \(|\hat{x}+\hat{y}|^2 + |\hat{y}+\hat{z}|^2 + |\hat{z}+\hat{x}|^2\).

In the following figure, \(AB\), \(DE\) and \(GF\) are parallel to each other and \(AD\), \(BG\) and \(EF\) are parallel to each other. If \(CD : CE = CG : CB = 2 : 1\), then the value of area \((\triangle AEG)\) : area \((\triangle ABD)\) is equal to

If a and b are two unit vectors, i.e., |a| = |b| = 1 and angle between them is \(\frac{\pi}{3}\), then \(16[\mathbf{a} \mathbf{b} + \mathbf{a} \times \mathbf{b} \mathbf{b}]\) is equal to:

If $\vec{a} = \hat{i}+2\hat{k}$, $\vec{b} = \hat{i}+\hat{j}+\hat{k}$, $\vec{c} = 7\hat{i}-3\hat{k}+4\hat{k}$, $\vec{r}\times\vec{b}+\vec{b}\times\vec{c} = \vec{0}$ and $\vec{r}\cdot\vec{a}=0$, then $\vec{r}\cdot\vec{c}$ is equal to

If $ABC$ be a triangle of sides $a, b, c$ with position vectors of $A, B, C$ as $\vec{a}, \vec{b}$ and $\vec{c}$ respectively, then the position vector of its incentre is:

Let $\vec{a}=2\hat{i}+3\hat{j}+4\hat{k}$, $\vec{b}=\hat{i}-2\hat{j}-2\hat{k}$, $\vec{c}=-\hat{i}+4\hat{j}+3\hat{k}$. If $\vec{d}\perp\vec{b},\vec{c}$ and $\vec{a}\cdot\vec{d}=18$, then $|\vec{a}\times\vec{d}|^2$ is equal to

If $\vec{a}$ is a unit vector and projection of $\vec{x}$ along $\vec{a}$ is 2 units and $(\vec{a} \times \vec{x}) \cdot \vec{b} = \vec{x}$, then $\vec{x}$ is given by:

If the vectors $\vec{a} = \lambda\hat{i} + \mu\hat{j} + 4\hat{k}$, $\vec{b} = 2\hat{i} + 4\hat{j} - 2\hat{k}$ and $\vec{c} = 2\hat{i} + 3\hat{j} + \hat{k}$ are coplanar and the projection of $\vec{a}$ on the vector $\vec{b}$ is $\sqrt{54}$ units, then the sum of all possible values of $\lambda + \mu$ is equal to

$\vec{a}=\hat{i}+4\hat{j}+2\hat{k}$, $\vec{b}=3\hat{i}-2\hat{j}+7\hat{k}$, $\vec{c}=2\hat{i}-\hat{j}+4\hat{k}$. $\vec{d}\times\vec{b}=\vec{c}\times\vec{b}$, $\vec{d}\cdot\vec{a}=24$. Then $|\vec{d}|^2$ is equal to