Find the value of \(\lambda\) for which the vectors \(\overrightarrow{a}=3\hat{i}+2\hat{j}+9\hat{k} \) and \(\overrightarrow{b}=\overrightarrow{i}+\lambda \overrightarrow{j}+3\overrightarrow{k}\) are parallel.

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

For any vector \(\overrightarrow{r}\) prove that \(\overrightarrow{r}\) = (\(\overrightarrow{r}.\hat{i}\)) \(\hat{i}\) + (\(\overrightarrow{r}.\hat{j}\)) \(\hat{j}\) + (\(\overrightarrow{r}.\hat{k}\)) \(\hat{k}\).

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

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 three vectors such that \(\overrightarrow{a}+2\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{0}\) and \(|\overrightarrow{a}|=3,|\overrightarrow{b}|=4,|\overrightarrow{c}|=7,\) find the angle between \(\overrightarrow{a}\) and \(\overrightarrow{b}\).

If \(|\overrightarrow{a}|=5,|\overrightarrow{b}|=6,|\overrightarrow{c}|=7\) and \(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c} =\overrightarrow{0}\), find \(\overrightarrow{a}.\overrightarrow{b}+\overrightarrow{b}.\overrightarrow{c}+\overrightarrow{c}.\overrightarrow{a}\).

Find the cosine and sine angle between the vectors \(\overrightarrow{a}=2\hat{i}+\hat{j}+3\hat{k}\) and  \(\overrightarrow{b}=4\hat{i}-2\hat{j}+2\hat{k}\).