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3-1.Vectors
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If the vectors $\vec P = a\hat i + a\hat j + 3\hat k$ and $\vec Q = a\hat i - 2\hat j - \hat k$ are perpendicular to each other then the positive value of $a$ is
A$0$
B$1$
C$2$
D$3$
Solution
$\overrightarrow{P} \cdot \overrightarrow{Q}=0$
(If $\overrightarrow{\mathrm{P}}$ and $\overrightarrow{\mathrm{Q}}$ are perpendicular to each other)
$(a \hat{i}+a \hat{j}+3 \hat{k}) \cdot(a \hat{i}-2 \hat{j}-\hat{k})=0$
$a \cdot a-a \cdot 2-3 \cdot 1=0$
$a^{2}-2 a-3=0$
$a^{2}-3 a+a-3=0$
$(a-3)(a+1)=0$
$a=3$ and $a=-1$
(If $\overrightarrow{\mathrm{P}}$ and $\overrightarrow{\mathrm{Q}}$ are perpendicular to each other)
$(a \hat{i}+a \hat{j}+3 \hat{k}) \cdot(a \hat{i}-2 \hat{j}-\hat{k})=0$
$a \cdot a-a \cdot 2-3 \cdot 1=0$
$a^{2}-2 a-3=0$
$a^{2}-3 a+a-3=0$
$(a-3)(a+1)=0$
$a=3$ and $a=-1$
Standard 11
Physics
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