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

A vector \(\overrightarrow{OP}\) makes 60° and 45° with the positive direction of the x and y axes respectively.  Then the angle between \(\overrightarrow{OP}\)and the z-axis is

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

The value of  \(\theta \in (0,{\pi\over 2})\) for which the vectors \(\overrightarrow{a}=(sin \theta)\hat{i}+(cos\theta)\hat{j}\) and \(\overrightarrow{b}=\hat{i}-\sqrt{3}\hat{j}+2\hat{k}\) are perpendicular, is equal to

If \(|\overrightarrow { a } |=|\overrightarrow { b } |\) then

If the direction cosines of a line are k, k and k, then

The value of \(\lambda\) when the vectors \(\vec{a}=2\vec{i}+\lambda\vec{j}+\vec{k}\) and \(​​\vec{b}=\vec{i}+2\vec{j}+3\vec{k}\) are orthogonal is ____________ .

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

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 \(​​​​\frac{\vec{a}+2 \vec{b}}{3} \text { and } \frac{\vec{b}+2 \vec{a}}{3} \text {. }\)

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

Verify whether the following ratios are direction cosines of some vector or not \({1\over\sqrt{2}},{1\over 2},{1\over 2}\)

Find  \(\overrightarrow{a}\).\(\overrightarrow{b}\)when \(\overrightarrow{a}=\hat{i}-2\hat{j}+\hat{k}\) and \(\overrightarrow{b}=3\hat{i}-4\hat{j}-2\hat{k}\)

Show that \(\overrightarrow{a}\times (\overrightarrow{b}+\overrightarrow{c})+\overrightarrow{b}\times (\overrightarrow{c}+\overrightarrow{a})+\overrightarrow{c}\times (\overrightarrow{a}+\overrightarrow{b})=\overrightarrow{0}\)

If \(\vec{a}=\hat{i}+2\hat{j}+3\hat{k}\) and \(\vec{b}=2\hat{i}+3\hat{j}-5\hat{k}\) then find \(\vec{a} \times \vec{b}\) . Verify that\(\vec{a}\) and \(\vec{b}\) are perpendicular to each other.

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

Find the angle between the vectors \(5\hat{i}+3\hat{j}+4\hat{k}\) and \(6\hat{i}-8\hat{j}-\hat{k}\).

If \((\overrightarrow { a } +\overrightarrow { b } ).(\overrightarrow { a } -\overrightarrow { b } )=0\) =0, then prove that \(\left| \overrightarrow { a } \right| =|\overrightarrow { b } |\)

If D is the midpoint of the side BC of a triangle ABC, prove that \(\overrightarrow{AB}\) + \(\overrightarrow{AC}\) = 2\(\overrightarrow{AD}\) 

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 \(\overrightarrow{EF}\). 

Show that the points (2, - 1, 3), (4, 3, 1) and (3, 1, 2) are collinear.