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.

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

If \(\overrightarrow{a}\) and \(\overrightarrow{b}\) represent a side and a diagonal of a parallelogram, find the other sides and the other diagonal.

Let A, B and C be the vertices of a triangle. Let D, E, and F be the midpoints of the sides BC, CA, and AB respectively. Show that \(\overrightarrow{AD}\) + \(\overrightarrow{BE}\) +\(\overrightarrow{CF}\) = \(\overrightarrow{0}\).

Show that the points whose position vectors are 2\(\hat{i}\) + 3\(\hat{j}\) − 5\(\hat{k}\), 3\(\hat{i}\) + \(\hat{j}\) − 2\(\hat{k}\) and, 6\(\hat{i}\) − 5\(\hat{j}\) + 7\(\hat{k}\) are collinear

Show that the points whose position vectors  4\(\hat{i}\) + 5\(\hat{j}\) + \(\hat{k}\) , -\(\hat{j}\) - \(\hat{k}\) , 3\(\hat{i}\) + 9\(\hat{j}\) + 4 \(\hat{k}\) ​and ​​​​​​ -4\(\hat{i}\) + 4\(\hat{j}\) + 4\(\hat{k}\) ​are coplanar.​​​​​​

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

If \(\overrightarrow{a},\overrightarrow{b},\)and \(\overrightarrow{c}\) are three unit vectors satisfying  \(\overrightarrow{a}-\sqrt{3}\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{0}\)  then find the angle between \(\overrightarrow{a}\) and \(\overrightarrow{c}\).

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 \(\overrightarrow{b}\).

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