Question

$\text{If}\overrightarrow{a}\text{and}\overrightarrow{b}$ are mutually perpendicular unit vectors, then $(3\overrightarrow{a}+2\overrightarrow{b})\cdot (5\overrightarrow{a}-6\overrightarrow{b})=$

• A. $5$
• B. $3$
• C. $6$
• D. $12$

Answer 1

The correct option is B $3$

$\overrightarrow{a}\text{and}\overrightarrow{b}$ are perpendicular.
$\Rightarrow \overrightarrow{a}\cdot \overrightarrow{b}=\overrightarrow{b}\cdot \overrightarrow{a}=0$
$\text{Also},|\overrightarrow{a}|=|\overrightarrow{b}|=1$
$\text{Now},(3\overrightarrow{a}+2\overrightarrow{b})\cdot (5\overrightarrow{a}-6\overrightarrow{b})$
$=3\overrightarrow{a}\cdot (5\overrightarrow{a}-6\overrightarrow{b})+2\overrightarrow{b}(5\overrightarrow{a}-6\overrightarrow{b})$
$=15|\overrightarrow{a}{|}^2-18\overrightarrow{a}\cdot \overrightarrow{b}+10\overrightarrow{b}\cdot \overrightarrow{a}-12|\overrightarrow{b}{|}^2$
$=15-12=3$