Vectors

If $\vec{a}, \vec{b}, \vec{c}$ are unit vectors, then $|\vec{a}-\vec{b}|^2 + |\vec{b}-\vec{c}|^2 + |\vec{c}-\vec{a}|^2$ does not exceed:

$ \vec{v} = 2\hat{i} + \hat{j} - \hat{k},\quad \vec{w} = \hat{i} + 3\hat{k} $ If $ \vec{u} $ is a unit vector, maximum value of $ [\vec{u}\ \vec{v}\ \vec{w}] $ is:

Let $\vec{a}, \vec{b}, \vec{c}$ be the position vectors of three vertices A, B, C of a triangle respectively then the area of this triangle is given by

For the vectors $\vec{a}=-4\hat{i}+2\hat{j}, \vec{b}=2\hat{i}+\hat{j}$ and $\vec{c}=2\hat{i}+3\hat{j}$, if $\vec{c}=m\vec{a}+n\vec{b}$ then the value of m + n is

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

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

Let $\vec{a}$ and $\vec{b}$ be two vectors, which of the following vectors are not perpendicular to each other?

The direction cosines of the vector a = (- 2i + j – 5k) are

A particle P starts from the point z₀=1+2i, where i=√−1 . It moves first horizontally away from origin by 5 units and then vertically away from origin by 3 units to reach a point z₁. From z₁ the particle moves √2 units in the direction of the vector $\hat{i}+\hat{j}$ and then it moves through an angle $\dfrac{\pi}{2}$ in anticlockwise direction on a circle with centre at origin, to reach a point z₂. The point z₂ is given by

If the volume of a parallelepiped whose adjacent edges are 

Vertices of the vectors i - 2j + 2k , 2i + j - k and 3i - j + 2k form a triangle. This triangle is

If a, b, c are three non-zero vectors with no two of which are collinear, a + 2b  is collinear with c and b + 3c is collinear with a , then | a + 2b + 6c | will be equal to

The position vectors of points A and B are  and  . Then the position vector of point p dividing AB in the ratio m : n is

If  are four vectors such that is collinear with  and is collinear with  then  =

If  are three non-coplanar vectors, then 

If $\vec{e_1}=(1,1,1)$ and $\vec{e_2}=(1,1,-1)$ and $\vec{a}$ and $\vec{b}$  and two vectors such that $\vec{e_2}=\vec{a}+2\vec{b}$ , then angle between $\vec{a}$ and $\vec{b}$

Angle between $\vec{a}$ and  $\vec{b}$ is $120{^{\circ}}$. If $|\vec{b}|=2|\vec{a}|$ and the vectors , $\vec{a}+x\vec{b}$ ,   $\vec{a}-\vec{b}$ are at right angle, then $x=$

Let $\vec{a}=\hat{i}+\hat{j}$ and  $\vec{b}=2\hat{i}-\hat{k}$, the point of intersection of the lines $\vec{r}\times\vec{a}=\vec{b}\times\vec{a}$  and  $\vec{r}\times\vec{b}=\vec{a}\times\vec{b}$  is

Let $\vec{a}=2\widehat{i}\, +\widehat{j}\, +2\widehat{k}$ , $\vec{b}=\widehat{i}-\widehat{j}+2\widehat{k}$ and $\vec{c}=\widehat{i}+\widehat{j}-2\widehat{k}$ are are three vectors. Then, a vector in the plane of $\vec{a}$ and $\vec{c}$ whose projection on $\vec{b}$ is of magnitude $\frac{1}{\sqrt{6}}$ is

If $\mathbf{a},\; \mathbf{b},\; \mathbf{c}$ are unit vectors such that $\mathbf{a} + \mathbf{b} + \mathbf{c} = 0$, then the value of $\mathbf{a}\cdot \mathbf{b} + \mathbf{b}\cdot \mathbf{c} + \mathbf{c}\cdot \mathbf{a}$ is:

The equation of the plane passing through the point $(1,2,3)$ and having the normal vector $N = 3\mathbf{i} - \mathbf{j} + 2\mathbf{k}$ is:

Let $\vec{A} = 2\hat{i} + \hat{j} – 2\hat{k}$ and $\vec{B} = \hat{i} + \hat{j}$, If $\vec{C}$ is a vector such that $|\vec{C} – \vec{A}| = 3$ and the angle between A × B and C is ${30^{\circ}}$, then $|(\vec{A} × \vec{B}) × \vec{C}|$ = 3 then the value of $\vec{A}.\vec{C}$ is equal to

If $\overrightarrow{{a}}$ and $\overrightarrow{{b}}$ are vectors in space, given by $\overrightarrow{{a}}=\frac{\hat{i}-2\hat{j}}{\sqrt[]{5}}$ and $\overrightarrow{{b}}=\frac{2\hat{i}+\hat{j}+3\hat{k}}{\sqrt[]{14}}$, then the value of$(2\vec{a} + \vec{b}).[(\vec{a} × \vec{b}) × (\vec{a} – 2\vec{b})]$ is

If a vector having magnitude of 5 units, makes equal angle with each of the three mutually perpendicular axes, then the sum of the magnitude of the projections on each of the axis is

$\theta={\cos }^{-1}\Bigg{(}\frac{3}{\sqrt[]{10}}\Bigg{)}$ is the angle between $\vec{a}=\hat{i}-2x\hat{j}+2y\hat{k}$ & $\vec{b}=x\hat{i}+\hat{j}+y\hat{k}$ then possible values of (x,y) that lie on the locus

If $\vec{a}, \vec{b}$ are unit vectors such that $2\vec{a}+\vec{b} =3$ then which of the following statement is true?

If $\vec{a}=\hat{i}-\hat{k}$, $\vec{b}=x\hat{i}+\hat{j}+(1-x)\hat{k}$ and $\vec{c}=y\hat{i}+x\hat{j}+(1+x-y)\hat{k}$, then $\begin{bmatrix}{\vec{a}} & {\vec{b}} & {\vec{c}}\end{bmatrix}$ depends on

The vector $\vec{A}=(2x+1)\hat{i}+(x^2-6y)\hat{j}+(xy^2+3z)\hat{k}$ is a

A man starts at the origin O and walks a distance of 3 units in the north- east direction and then walks a distance of 4 units in the north-west direction to reach the point P. then $\vec{OP}$ is equal to

Let $\vec{a}=2\hat{i}-3\hat{j}+4\hat{k}$, $\vec{b}=\hat{i}+2\hat{j}-\hat{k}$ and $\vec{c}=3\hat{i}+\hat{j}+\lambda \hat{k}$ be the co-terminal edges

Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be unit vectors such that the angle between them is ${\cos }^{-1}\Bigg{\{}\frac{1}{4}\Bigg{\}}$. If $\vec{b}=2\vec{c}+\lambda \vec{a}$, where $\lambda$ > 0 and $\vec{b}=4$, then $\lambda$ is equal to

The length of the projection of $\vec{a} = 2\hat{i} + 3\hat{j} + \hat{k}$ on $\vec{b} = -2\hat{i} + \hat{j} + 2\hat{k}$, is equal to: