If D and E are the midpoints of the sides AB and AC of a triangle ABC, prove that \(\overrightarrow{BE}+\overrightarrow{DC}={3\over2}\overrightarrow{BC}\) 

Find a unit vector along the direction of the vector 5\(\hat{i}\) - 3\(\hat{j}\) + 4\(\hat{k}\) .

Find the direction cosines of the line joining (2, 3, 1) and (3, - 1, 2).

Verify whether the following ratios are direction cosines of some vector or not \({4\over 3},0,{3\over 4}\)

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

If \(\overrightarrow{a}\) and \(\overrightarrow{b}\)are two vectors such that | \(\overrightarrow{a}\) | = 10, | \(\overrightarrow{b}\) | = 15 and \(\overrightarrow{a}\).\(\overrightarrow{b}\) = 75 \(\sqrt{2}\), find the angle between \(\overrightarrow{a}\) and \(\overrightarrow{b}\).

Find the angle between the vectors  \(2\hat{i}+3\hat{j}-6\hat{k}\) and \(6\hat{i}-3\hat{j}+2\hat{k}\)

Find the area of the parallelogram whose adjacent sides are \(\overrightarrow{a}=3\hat{i}+\hat{j}+4\hat{k}\) and \(\overrightarrow{b}=\hat{i}-\hat{j}+\hat{k}\).

Classify the following as scalar and vector quantities. (i) time period (ii) distance (iii) force (iv) velocity (v) work done.

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.