Question
Download Solution PDFA point on a line has coordinates (p + 1, p - 3, √2p) where p is any real number. What are the direction cosines of the line?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
If a, b and c are direction ratio’s of a line then direction cosines of the line are given by:
\(l=\frac{a}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}},~m=~\frac{b}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}}~and~n=~\frac{c}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}}\)
Calculation:
Given: A point on a line has coordinates (p + 1, p - 3, √2p)
⇒ x = p + 1 ⇒ x - 1 = p ---(1)
⇒ y = p - 3 ⇒ y + 3 = p ---(2)
⇒ z = √2 × p ⇒ \(\frac{z}{\sqrt{2}}\) = p ---(3)
From (1), (2) and (3) we can say that
\(\frac{x-1}{1}=\frac{y+3}{1}=\frac{z-0}{\sqrt{2}}=p\)
∴ Direction ratios are: 1, 1, √2
⇒ Direction cosines are: \(\frac{1}{2},\frac{1}{2},\frac{1}{\sqrt{2}}\)Last updated on May 30, 2025
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