Answer to Question #97564 in Analytic Geometry for Ojugbele Daniel

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