Vector Algebra Questions (573)

If O is origin and C is the mid-point of A(2, -1) and B(-4, 3), then the value of OC is

Let $OPQR$ is a tetrahedon such that O is origin and $\vec{p}, \vec{q}, \vec{r}$ are position vectors of P, Q, R respectively and $\alpha$ is the angle which OP makes with face PQR then:

Which of the following statements is/are correct? (a) If \(\mathbf{n} \cdot \mathbf{a} = 0, \mathbf{n} \cdot \mathbf{b} = 0\) and \(\mathbf{n} \cdot \mathbf{c} = 0\) for some non-zero vector \(\mathbf{n}\), then \([\mathbf{a} \mathbf{b} \mathbf{c}] = 0\)

Let \(\vec{a}\), \(\vec{b}\), \(\vec{c}\) be distinct non-negative numbers. If the vectors \(a\vec{i} + a\vec{j} + c\vec{k}\), \(\vec{i} + \vec{k}\) and \(c\vec{i} + c\vec{j} + b\vec{k}\) lie in a plane, then \(c\) is

If the interior and exterior bisectors of the angle $A$ of a triangle $ABC$ meet the side $BC$ at $D$ and $E$, then :

Let OABCD be a pentagon in which the sides OA and CB are parallel and the sides OD and AB are parallel. Also, OA : CB = 2 : 1 and OD : AB = 1 : 3. The ratio \(\frac{AX}{XD}\) is

The direction cosines of vector a = 3i + 4j + 5k in the direction of positive axis of X, is

Ex. 64: Find the least positive integral value of x for which the angle between vectors \(\vec{a} = x\vec{i} - 3\vec{j} - \vec{k}\) and \(\vec{b} = 2x\vec{i} + x\vec{j} - \vec{k}\) is acute.

The resultant displacement of the ant after first two steps is:

The vector, directed along the internal bisector of the angle between the vectors $\vec{a} = 7\vec{i} - 4\vec{j} - 4\vec{k}$ & $\vec{b} = -2\vec{i} - \vec{j} + 2\vec{k}$ with $|\vec{c}| = 5\sqrt{6}$ is:

Ex. 54 The ratio \(PQ : DB\) is equal to

Ex. 106: Find AN + BP + CM where points M, N and P are taken on sides AB, BC and CA respectively, such that \frac{AM}{AB} = \frac{BN}{BC} = \frac{CP}{CA} = \alpha, with A at the origin, B and C having position vectors b and c respectively.

OP · OQ, where O is the origin (P is on the circle and curve, Q is inside the circle with integer abscissa)

Let position vectors of points A, B and C of triangle ABC respectively be \(\vec{i} + \vec{j} + 2\vec{k}\), \(\vec{i} + 2\vec{j} + \vec{k}\) and \(2\vec{i} + \vec{j} + \vec{k}\). Let \(l_1\), \(l_2\) and \(l_3\) be the lengths of perpendiculars drawn from the orthocenter 'O' on the sides AB, BC and CA, then \((l_1 + l_2 + l_3)\) equals

Let a vector $\vec{a}=\sqrt{2}\hat{i}-\hat{j}+\lambda\hat{k}$, $\lambda>0$, make an obtuse angle with the vector $\vec{b}=-\lambda^2\hat{i}+4\sqrt{2}\hat{j}+4\sqrt{2}\hat{k}$ and an angle $\theta$, $\dfrac{\pi}{6}<\theta<\dfrac{\pi}{2}$, with the positive $z$-axis. If the set of all possible values of $\lambda$ is $(\alpha,\beta)-\{\gamma\}$, then $\alpha+\beta+\gamma$ is equal to ___.

\((\vec{a} \cdot \vec{i})\vec{i} + (\vec{a} \cdot \vec{j})\vec{j} + (\vec{a} \cdot \vec{k})\vec{k}\) is equal to

If a + b + c = 0 and |a| = 3, |b| = 5, |c| = 7, then the angle between a and b is

The direction cosines of the vector 3i - 4j + 5k are

In a triangle AOB, R and Q are the points on the side OB and AB respectively such that \(3OR = 2RB\) and \(2AQ = 3QB\). Let OQ and AR intersect at the point P (where O is origin). If the point P divides OQ in the ratio of m:1, then m is:

A vector equally inclined to the vectors \(\mathbf{i} - \mathbf{j} + \mathbf{k}\) and \(\mathbf{i} + \mathbf{j} - \mathbf{k}\) then the plane containing them is

B divided AC in ratio

Let $\vec{u}$ and $\vec{v}$ be unit vectors. if $\vec{w}$ is a vector such that $\vec{w} + (\vec{w} \times \vec{u}) = \vec{v}$, if the maximum volume of the parallelepiped formed by $\vec{u}, \vec{v}$ and $\vec{w}$ is $p$ then $12p = _______.

Given a parallelogram ABCD. If |AB| = a, |AD| = b and |AC| = c, then DB · AB has the value

Five points given by A, B, C, D and E are in a plane. Three forces $\vec{AC}$, $\vec{AD}$ and $\vec{AE}$ act at A and three forces $\vec{CB}$, $\vec{DB}$ and $\vec{EB}$ act at B. Then, their resultant is

If the lines $\vec{r} = \vec{a} + (\vec{b} \times \vec{c})$, and $\vec{r} = \vec{b} + s(\vec{c} \times \vec{a})$ intersect (t and s are scalars) then:

Ex. 99 Which of the following is true?

Let $A(2,3,5)$ and $C(-3,4,-2)$ be opposite vertices of a parallelogram $ABCD$. If the diagonal $\overrightarrow{BD}=\hat{i}+2\hat{j}+3\hat{k}$, then the area of the parallelogram is equal to:

[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

If OABC is a tetrahedron such that $OA^2 + BC^2 = OB^2 + CA^2 = OC^2 + AB^2$, then which of the following is not true?

Let \(\vec{a} = \vec{i} + 2\vec{j} - 3\vec{k}\) and \(\vec{b} = 2\vec{i} - 3\vec{j} + 5\vec{k}\). If \(\vec{r} \times \vec{a} = \vec{b} \times \vec{r}\), \(\vec{r} \times (a\vec{i} + 2\vec{j} + \vec{k}) = 3\) and \(\vec{r} \times (2\vec{i} + 5\vec{j} - a\vec{k}) = -1\), where \(a \in \mathbb{R}\), then the value of \(a + |\vec{r}|^2\) is equal to

If b and c are orthogonal unit vectors and \(\mathbf{b} \times \mathbf{c} = \mathbf{a}\), then \([\mathbf{a} + \mathbf{b} + \mathbf{c} \mathbf{a} + \mathbf{b} \mathbf{b} + \mathbf{c}]\) is equal to:

$p_1 + p_2$ is equal to:

If the non-zero vectors $\vec{a}$ and $\vec{b}$ are perpendiculars to each other then the solution of the equation $\vec{r} \times \vec{a} = \vec{b}$ is given by:

The position vectors of vertices of ∆ABC are \(a\), \(b\), \(c\) and \(a \cdot a = b \cdot b = c \cdot c = 3\). If \([a b c] = 0\), then the position vector of the orthocentre of ∆ABC is

A unit vector perpendicular to the vector $\mathbf{i} + 2\mathbf{j} + 2\mathbf{k}$ and making equal angles with X and Y-axes can be:

If $4\vec{a} + 5\vec{b} + 9\vec{c} = \vec{0}$, then $(\vec{a} \times \vec{b}) \times (\vec{b} \times \vec{c}) \times (\vec{c} \times \vec{a})$ is equal to:

Let three vectors \(\vec{a},\vec{b},\vec{c}\) satisfy \(\vec{a}\times\vec{b}=\vec{c}\) and \(\vec{b}\times\vec{c}=\vec{a}\), with \(|\vec{a}|=1\). If the angle between \(\vec{b}\) and \(\vec{c}\) is \(\dfrac{\pi}{6}\), find \(|\vec{b}|\).

If a and b are the vectors determined by two adjacent sides of a regular hexagon, then vector EO is

Let OABCD be a pentagon in which the sides OA and CB are parallel and the sides OD and AB are parallel. Also, OA : CB = 2 : 1 and OD : AB = 1 : 3. The ratio \(\frac{OX}{XC}\) is

A vector whose modulus is \(\sqrt{51}\) and makes the same angle with a = \(\frac{\mathbf{i} - 2\mathbf{j} + 2\mathbf{k}}{3}\), b = \(\frac{-4\mathbf{i} - 3\mathbf{k}}{5}\) and c = j, will be

If the ratio of area of quadrilateral PQBR and area of △OPA is \frac{a}{b}, then find (b − a) where a and b are coprime numbers.

The 3-dimensional vectors v1, v2, v3 satisfying \(\mathbf{v}_1 \cdot \mathbf{v}_1 = 4\), \(\mathbf{v}_1 \cdot \mathbf{v}_2 = -2\), \(\mathbf{v}_1 \cdot \mathbf{v}_3 = 6\), \(\mathbf{v}_2 \cdot \mathbf{v}_2 = 2\), \(\mathbf{v}_2 \cdot \mathbf{v}_3 = -5\), \(\mathbf{v}_3 \cdot \mathbf{v}_3 = 29\), then v3 may be

Let \(\vec{a}=2\hat{i}-\hat{j}+4\hat{k}\) and \(\vec{b}=\hat{i}+\alpha\hat{j}+\beta\hat{k}\). If \(\vec{b}\) is perpendicular to \(3\hat{i}-4\hat{j}+\hat{k}\) and the projection of \(\vec{b}\) on \(\vec{a}\) is \(\dfrac{17}{\sqrt{21}}\), find \(|\vec{b}|\).

A(2, 6, 2), B(-4, 0, $\lambda$), C(2, 3, -1) and D(4, 5, 0), $|\lambda| \leq 5$ are the vertices of a quadrilateral ABCD. If its area is 18 square units, then $5-6\lambda$ is equal to _____.

Let $\vec{v} = \alpha\hat{i}+2\hat{j}-3\hat{k}$, $\vec{w} = 2\alpha\hat{i}+\hat{j}-\hat{k}$, and $\vec{u}$ be a vector such that $|\vec{u}|=\alpha>0$. If the minimum value of the scalar triple product $[\vec{u}\,\vec{v}\,\vec{w}]$ is $-\alpha\sqrt{3401}$, and $|\vec{u}\cdot\hat{i}|^2 = \frac{m}{n}$ where m and n are coprime natural numbers, then $m+n$ is equal to _____.

Two given points $P$ and $Q$ in the rectangular cartesian coordinates lie on $y = 2^{x^2}$ such that $\overrightarrow{OP} \cdot \hat{i} = -1$ and $\overrightarrow{OQ} \cdot \hat{i} = +2$ where $\hat{i}$ is a unit vector along the x-axis. The magnitude of $\frac{\overrightarrow{OQ} - 4\overrightarrow{OP}}{2}$ is _______.

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{a}|=\sqrt{14}$, $|\vec{b}|=\sqrt{6}$ and $|\vec{a}\times\vec{b}|=\sqrt{48}$. Then $(\vec{a}\cdot\vec{b})^2$ is equal to _____.

For two particular vectors A and B, it is known that A × B = B × A. What must be true about the two vectors?

Arc $PQ$ subtends right angle at centre $O$. Midpoint $R$ of arc. $\overrightarrow{OP}=\vec{u}$, $\overrightarrow{OR}=\vec{v}$, $\overrightarrow{OQ}=\alpha\vec{u}+\beta\vec{v}$. Then $\alpha,\beta^2$ satisfy

$\vec{a}=\hat{i}+2\hat{j}+3\hat{k}$, $\vec{b}=\hat{i}+\hat{j}-\hat{k}$. $\vec{c}$: $\vec{a}\cdot\vec{c}=11$, $\vec{b}\cdot(\vec{a}\times\vec{c})=27$, $\vec{b}\cdot\vec{c}=-\sqrt{3}|\vec{b}|$. Then $|\vec{a}\times\vec{c}|^2$ is equal to