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

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

Find \(\overrightarrow{a}.\overrightarrow{b}\) when \(\overrightarrow{a}\)= \(\hat{i}-\hat{j}+5\hat{k}\)  and \(\overrightarrow{b}=3\hat{i}-2\hat{k}\)

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 \(|\overrightarrow{a}+\overrightarrow{b}|=|\overrightarrow{a}-\overrightarrow{b}|\) prove that \(\overrightarrow{a}\) and \(\overrightarrow{b}\) are perpendicular.

Find the value \(\lambda\) for which the vectors \(\overrightarrow{a}\) and \(\overrightarrow{b}\) are perpendicular, where \(\overrightarrow{a}=2\hat{i}+\lambda\hat{j}+\hat{k}\) and \(\overrightarrow{b}=\hat{i}-2\hat{j}+3\hat{k}\)

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

Find the angle between the vectors \(2\overrightarrow{i}+\overrightarrow{j}-\overrightarrow{k}\) and \(\overrightarrow{i}+2\overrightarrow{j}+\overrightarrow{k}\) using vector product

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

If \(\vec{a}=\hat{i}+2\hat{j}+3\hat{k}\) , \(\vec{b}=-\hat{j}+4\hat{k}\)and \(\vec{c}=\hat{i}-2\hat{j}+\hat{k}\). Find\((\vec{a}+\vec{b}).\vec{c}\) and \(\vec{c}.(\vec{a}+\vec{b})\) Are they equal?