Question

Find b so that the vectors $v=i+j$ and $w=i+bj$ are orthogonal.

Short answer

The value of$b$is$-1$.

Step-by-step solution

Step 1. Concept Used 

When two vectors uand vare orthogonal then their dot product is zero that is $u·v=0$.

Step 2. Find the dot product 

For the vectors $v=i+j$and $w=i+bj$the dot product is given as

$$\begin{gathered} v·w=(i+j)·(i+bj) \\ v·w=1·1+1·b \\ v·w=1+b \end{gathered}$$

Step 3. Form the equation and solve for b

As the vectors $v$and $w$are orthogonal so their dot product is zero and so the equation is formed as

$\begin{array}{rcl}v·w& =& 0\\ 1+b& =& 0\end{array}$

On solving the equation we get

$\begin{array}{rcl}1+b-1& =& 0-1\\ b& =& -1\end{array}$