Prove that the line segments joining the midpoints of the adjacent sides of a quadrilateral form a parallelogram.

Prove that the points whose position vectors \(2\hat{i}+4\hat{j}+3\hat{k},4\hat{i}+\hat{j}+9\hat{k}\) and \(10\hat{i}-\hat{j}+6\hat{k}\) form a right angled triangle.

Show that the vectors \(5\hat{i}+6\hat{j}+7\hat{k},7\hat{i}-8\hat{j}+9\hat{k},3\hat{i}+20\hat{j}+5\hat{k}\) are coplanar.

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\(\hat{k}\) .

Let \(\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\) be three vectors such that | \(\overrightarrow{a}\) | = 3, | \(\overrightarrow{b}\) | = 4, | \(\overrightarrow{c}\) | = 5 and each one of them being perpendicular to the sum of the other two, find | \(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}\) |.

Prove that the smallar angle between any two diagonals of a cube is cos^-1 \(({1\over3})\).

Let A, Band C represent the angles of a \(\triangle\)ABC and a, band c represent the lengths of the sides opposite to them, then prove that a² = b² + c² - 2bc cos A (Law of cosines)

Let \(\overrightarrow { a } =\hat { i } +\hat { j } +2\hat { k } \) and \(\overrightarrow { b } =\hat { i } +2\hat { j } +\hat { k } \) and \(\overrightarrow { c } \)  be a unit vectorin the plane determined by \(\overrightarrow { a } \) and \(\overrightarrow { b } \). If \(\overrightarrow { c } \) is perpendicular to the vector \(\hat { i } +\hat { j } +\hat { k } \) and makes an obtuse angle with \(\overrightarrow { a } \), then prove that \(\overrightarrow { c } =\frac { \hat { j } -\hat { k } }{ \sqrt { 2 } } \)

Let A, Band C represent the angles of a \(\triangle\)ABC and a, b, c represent the lengths of the sides opposite to them, then prove that a = b cos C + c cos B (Projection formula)