Question

A vector v has initial point $P=\left(1,3,-2\right)$and terminal point $Q=\left(4,-2,1\right).$ Write vin the form$v=ai+bj+ck.$

Short answer

The vector is$3i-5j+3k.$

Step-by-step solution

Step 1. Given Information

The initial point of vector v is$P=\left(1,3,-2\right)$and the terminal point of vector v is$Q=\left(4,-2,1\right).$

Step 2. Writing vector v in the form v=ai+bj+ck

To write the vector v, we will use initial and terminal points in the way such as $v=\left(x_2-x_1\right)\mathit{i}+\left(y_2-y_1\right)\mathit{j}+\left(z_2-z_1\right)\mathit{k}$

Here,

$$\begin{gathered} P=\left(x_1,y_1,z_1\right)=\left(1,3,-2\right) \\ Q=\left(x_2,y_2,z_2\right)=\left(4,-2,1\right) \end{gathered}$$

So,

role="math" localid="1646990954182" $$\begin{gathered} v=\left(x_2-x_1\right)\mathbf{i}+\left(y_2-y_1\right)\mathbf{j}+\left(z_2-z_1\right)\mathbf{k} \\ v=\left(4-1\right)\mathbf{i}+\left(-2-3\right)\mathbf{j}\mathbf{}+\left(1+2\right)\mathbf{k} \\ \mathrm{v}=3\mathbf{i}-5\mathbf{j}+3\mathbf{k} \end{gathered}$$