Question

If the sum of three vectors in R is zero, must they lie in the same plane? Explain.

Answer 1

Answer with Step-by-step explanation:

We are given that sum of three vectors in
`R^3` is zero.

We have to tell and explain that if sum of three vectors in
`R^3` is zero then they must lie in the same plane or not.

We know that if three or more vectors lie in the same then they are coplanar.

If three vectors are co-planar then their scalar product is zero.

According to given condition

Let (x,0,0), (-x,0,0) and (0,0,0)

`\vec{u}=x\hat{i}`

`\vec{v}=-x\hat{i}`

`\vec{w}=0`

Sum of three vectors=
`(x-x+0)\hat{i}+(0+0+)\hat{j}+(0+0+0)\hat{k}=0`

`u\cdot (v* w)=\begin{vmatrix}x_1&y_1&z_1\\x_2&y_2&z_2\\x_3&y_3&z_3\end{vmatrix}`

`u\cdot (v* w)=\begin{vmatrix}x&0&0\\-x&0&0\\0&0&0\end{vmatrix}`

When all elements of one row or column are zero of square matrix A then det(A)=0

`u\cdot (v* w)=0`

Therefore, vectors u,v and w are co-planar.

Hence, if the sum of three vectors in
`R^3` is zero then they must lie in the same plane.