Question

Vector a= 2i+2j+3k
Vector b=i+2j+k
Vector c= 3i+j
Are such that a+yb is perpendicular to c find y.

Answer 1

Find the vector $y\overrightarrow{b}$:

$$y\overrightarrow{b}=y\overrightarrow{i}+2y\overrightarrow{j}+y\overrightarrow{k}$$

Tnen

$$\overrightarrow{a}+y\overrightarrow{b}=(2\overrightarrow{i}+2\overrightarrow{j}+3\overrightarrow{k})+(y\overrightarrow{i}+2y\overrightarrow{j}+y\overrightarrow{k})$$ $$\overrightarrow{a}+y\overrightarrow{b}=(2+y)\overrightarrow{i}+(2+2y)\overrightarrow{j}+(3+y)\overrightarrow{k}$$

If $\overrightarrow{a}+y\overrightarrow{b}$ is perpendicular to $\overrightarrow{c}$ , then their dot product is equal to zero:

$$(2+y)\cdot 3+(2+2y)\cdot 1+(3+y)\cdot 0=0$$

Simplify:

$$6+3y+2+2y=0$$ $$5y+8=0$$ $$y=-\frac{8}{5}$$ $$y=-1.6$$

Answer: -1.6