The value of \(\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{DA}+\overrightarrow{CD}\) is

The unit vector parallel to the resultant of the vectors \(\hat{i}+\hat{j}-\hat{k}\) and \(\hat{i}-2\hat{j}+\hat{k}\) is

The vectors \(\overrightarrow{a}-\overrightarrow{b},\overrightarrow{b}-\overrightarrow{c},\overrightarrow{c}-\overrightarrow{a}\) are

One of the diagonals of parallelogram ABCD with \(\overrightarrow{a}\) and \(\overrightarrow{b}\) as adjacent sides is \(\overrightarrow{a}+\overrightarrow{b}\) The other diagonal \(\overrightarrow{BD}\) is

If \(\overrightarrow{a},\overrightarrow{b}\) are the position vectors A and B, then which one of the following points whose position vector lies on AB, is

Two vertices of a triangle have position vectors \(3\hat{i}+4\hat{j}-4\hat{k}\) and \(2\hat{i}+3\hat{j}+4\hat{k}\) . If the position vector of the centroid is \(\hat{i}+2\hat{j}+3\hat{k}\), then the position vector of the third vertex is

If \(\overrightarrow{a}\)  and \(\overrightarrow{b}\) having same magnitude and angle between them is 60° and their scalar product is \({1\over2}\) then \(|\overrightarrow{a}|\) is 

If   \(\overrightarrow{a}\)  and   \(\overrightarrow{b}\) are two vectors of magnitude 2 and inclined at an angle 60°, then the angle between   \(\overrightarrow{a}\)  and \(\overrightarrow{a}+\overrightarrow{b}\) is

If the projection of \(5\hat{i}-\hat{j}-3\hat{k}\) on the vector \(\hat{i}+3\hat{j}+\lambda\hat{k}\) is same as the projection of \(\hat{i}+3\hat{j}+\lambda\hat{k}\) on \(5\hat{i}-\hat{j}-3\hat{k}\), then \(\lambda\) is equal to

If \(\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k},\overrightarrow{b}=2\hat{i}+x\hat{j}+\hat{k},\overrightarrow{c}=\hat{i}-\hat{j}+4\hat{k}\) and \(\overrightarrow{a}.(\overrightarrow{b}\times \overrightarrow{c})=70,\) then x is equal to

Represent graphically the displacement of 80km, 60° south of west.

Let \(\vec a\) and \(\vec b\) be the position vectors of the points A and B. Prove that the position vectors of the points which trisects the line segment AB are \(​​​​\frac{\vec{a}+2 \vec{b}}{3} \text { and } \frac{\vec{b}+2 \vec{a}}{3} \text {. }\)

Find a direction ratio and direction cosines of the following vectors \(3\hat{i}+4\hat{j}-6\hat{k}\)

Can a vector have direction angles 30°, 45°, 60°?

The direction ratios of a vector are 2, 3, 6 and it’s magnitude is 5. Find the vector

Find the direction cosines of a vector whose direction ratios are 1, 2, 3

Find the angle between the vectors \(\hat{i}-\hat{j}\) and \(\hat{j}-\hat{k}\).

Find the vectors of magnitude \(10\sqrt{3}\) that are perpendicular to the plane which contains \(\hat{i}+2\hat{j}+\hat{k}\) and\(\hat{i}+3\hat{j}+4\hat{k}\)

Find the unit vectors perpendicular to each of the vectors \(\overrightarrow{a}+\overrightarrow{b}\) and \(\overrightarrow{a}-\overrightarrow{b}\), where \(\overrightarrow{a}=\hat{i}+\hat{j} +\hat{k} \) and \(\overrightarrow{b} =\hat{i}+2\hat{j} +3\hat{k} \).

If \(\overrightarrow{PO}\) +\(\overrightarrow{OQ}\) = \(\overrightarrow{QO}\) +\(\overrightarrow{OR}\), prove that the points P, Q, R are collinear.

Show that the points whose position vectors are 2\(\hat{i}\) + 3\(\hat{j}\) − 5\(\hat{k}\), 3\(\hat{i}\) + \(\hat{j}\) − 2\(\hat{k}\) and, 6\(\hat{i}\) − 5\(\hat{j}\) + 7\(\hat{k}\) are collinear

If (a, a + b, a + b + c) is one set of direction ratios of the line joining (1, 0, 0) and (0, 1, 0), then find a set of values of a, b, c.

Show that the following vectors are coplanar 5\(\hat{i}\) +6\(\hat{j}\) +7\(\hat{k}\) ,7 \(\hat{i}\) -8\(\hat{j}\) +9 \(\hat{k}\),3\(\hat{i}\)+20\(\hat{j}\) +5\(\hat{k}\) .

If \(\overrightarrow{a}=2\hat{i}+2\hat{j}+3\hat{k},\) \(\overrightarrow{b}=-\hat{i}+2\hat{j}+\hat{k}\) and \(\overrightarrow{c}=3\hat{i}+\hat{j}\) be such that \(\overrightarrow{a}+\lambda \overrightarrow{b}\) is perpendicular to \(\overrightarrow{c}\) then find \(\lambda\).

If \(\overrightarrow{a},\overrightarrow{b}\) are unit vectors and \(\theta\) is the angle between them, show that \(cos {\theta \over 2}={1\over2}|\overrightarrow{a}+\overrightarrow{b}|\)

If \(\overrightarrow{a},\overrightarrow{b}\) are unit vectors and \(\theta\) is the angle between them, show that \(tan {\theta \over 2}={|\overrightarrow{a}-\overrightarrow{b}|\over|\overrightarrow{a}+\overrightarrow{b}|}\)

Three vectors \(\overrightarrow{a},\overrightarrow{b}\)and \(\overrightarrow{c}\) are such that \(|\overrightarrow{a}|=2,|\overrightarrow{b}|=3,|\overrightarrow{c}|=4,\) and \(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{0}\) .Find \(4\overrightarrow{a}.\overrightarrow{b}+​​3\overrightarrow{b}.\overrightarrow{c}+3\overrightarrow{c}.\overrightarrow{a}.\)