MHT CET · Maths · Vector Algebra
Scalar projection of the line segment joining the points \(\mathrm{A}(-2,0,3), \mathrm{B}(1,4,2)\) on the line whose direction ratios are \(6,-2,3\) is
- A \(\frac{23}{7}\)
- B 1
- C 7
- D \(\frac{1}{7}\)
Answer & Solution
Correct Answer
(B) 1
Step-by-step Solution
Detailed explanation
Let \(\bar{a}\) be the vector joining \(\mathrm{A}(-2,0,3)\) and \(\mathrm{B}(1,4,2)\).
\(\begin{aligned}
& \therefore \quad \overline{\mathrm{a}}=(1-(-2)) \hat{\mathrm{i}}+(4-0) \hat{\mathrm{j}}+(2-3) \hat{\mathrm{k}} \\
& =3 \hat{i}+4 \hat{j}-\hat{k} \\
& \text { and } \bar{b}=6 \hat{i}-2 \hat{j}+3 \hat{k} \\
& \therefore \quad \text { Projection }=\frac{\overline{\mathrm{a}} \cdot \overline{\mathrm{b}}}{|\overline{\mathrm{b}}|}=\frac{3 \times 6+4 \times(-2)-1 \times 3}{\sqrt{6^2+(-2)^2+3^2}} \\
& =\frac{18-8-3}{\sqrt{49}} \\
& =\frac{7}{7} \\
& =1 \\
&
\end{aligned}\)
\(\begin{aligned}
& \therefore \quad \overline{\mathrm{a}}=(1-(-2)) \hat{\mathrm{i}}+(4-0) \hat{\mathrm{j}}+(2-3) \hat{\mathrm{k}} \\
& =3 \hat{i}+4 \hat{j}-\hat{k} \\
& \text { and } \bar{b}=6 \hat{i}-2 \hat{j}+3 \hat{k} \\
& \therefore \quad \text { Projection }=\frac{\overline{\mathrm{a}} \cdot \overline{\mathrm{b}}}{|\overline{\mathrm{b}}|}=\frac{3 \times 6+4 \times(-2)-1 \times 3}{\sqrt{6^2+(-2)^2+3^2}} \\
& =\frac{18-8-3}{\sqrt{49}} \\
& =\frac{7}{7} \\
& =1 \\
&
\end{aligned}\)
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