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

If \(\overrightarrow{a}+2\overrightarrow{b}\) and \(3\overrightarrow{a}+m\overrightarrow{b}\) are parallel, then the value of m is

If \(\overrightarrow{r}={9\overrightarrow{a}+7\overrightarrow{b}\over16}\), then the point P whose position vector \(\overrightarrow{r}\) divides the line joining the points with position vectors \(\overrightarrow{a}\) and \(\overrightarrow{b}\) in the ratio

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 \(|\overrightarrow{a}|=13,|\overrightarrow{b}|=5\)  and \(\overrightarrow{a}.\overrightarrow{b}=60^o\) then \(|\overrightarrow{a}\times\overrightarrow{b}|\) is

Represent graphically the displacement of (i) 30 km 60° west of north (ii) 60 km 50° south of east.

Prove that the relation R defined on the set V of all vectors by ‘ \(\overrightarrow{a}\ R\ \overrightarrow{b} \ if \ \overrightarrow{a}=\overrightarrow{b}\) is an equivalence relation on V.

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

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

Find the direction cosines and direction ratios for the following vectors. 5\(\hat{i}\) - 3\(\hat{j}\) - 48\(\hat{k}\)

Let A and B be two points with position vectors 2\(\overrightarrow{a}\)+ 4\(\overrightarrow{b}\) and 2\(\overrightarrow{a}\) − 8\(\overrightarrow{b}\). Find the position vectors of the points which divide the line segment joining A and B in the ratio 1:3 internally and externally.

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

Show that the following vectors are coplanar \(\hat{i}\) − 2\(\hat{j}\) + 3\(\hat{k}\), - 2\(\hat{i}\) + 3\(\hat{j}\) - 4\(\hat{k}\) ,-\(\hat{j}\) + 2\(\hat{k}\) .

Show that the points (4, - 3, 1), (2, - 4, 5) and (1, - 1, 0) form a right angled triangle.

If \(\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\)are position vectors of the vertices A, B, C of a triangle ABC, show that the area of the triangle ABC is \({1\over 2}|\overrightarrow{a}\times \overrightarrow{b}+ \overrightarrow{b}+\overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a}|\). Also deduce the condition for collinearity of the points A, B, and C.

If ABCD is a quadrilateral and E and F are the midpoints of AC and BD respectively, then prove that \(\overrightarrow{AB}\) + \(\overrightarrow{AD}\) + \(\overrightarrow{CB}\) +\(\overrightarrow{CD}\) = 4 \(\overrightarrow{EF}\). 

The position vectors of the points P, Q, R, S are \(\hat{i}\) + \(\hat{j}\) + \(\hat{k}\), 2 \(\hat{i}\) + 5\(\hat{j}\), 3\(\hat{i}\) + 2\(\hat{j}\) - 3\(\hat{k}\), and \(\hat{i}\) - 6\(\hat{j}\) - \(\hat{k}\), respectively. Prove that the line PQ and RS are parallel.

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

Find the projection of the vector \(\hat{i}+3\hat{j}+7\hat{k}\) on the vector\(2\hat{i}+6\hat{j}+3\hat{k}\).