CBSE 2024 · Region 5 · Set 1 · Q25 · 2 marks
Find the vector equation of the line passing through the point $\displaystyle (2,3,-5)$ and making equal angles with the co-ordinate axes.
Marking-scheme solution
(a)
$\displaystyle \cos \alpha=\cos \beta=\cos \gamma=l \Rightarrow l^{2}+l^{2}+l^{2}=1 \Rightarrow 3 l^{2}=1, \therefore l=1 / \sqrt{3}$
Direction cosines of the line are $\displaystyle \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \Rightarrow$ the direction ratios are $\displaystyle 1,1,1$
∴ Vector equation of the line is: $\displaystyle \overrightarrow{\mathbf{r}}=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-5 \hat{\mathbf{k}}+\lambda(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}})$
Three Dimensional GeometryDirection Cosines and Direction Ratios of a LineApplyvery_short_answermedium
More from Three Dimensional Geometry
- Find the image A^′ of the point A(1,6,3) in the line x/1=(y-1)/2=(z-2)/3. Also, find the equation of the line…2025 · asked 3×
- Find the values of λ, for which the distance of point (2,1, λ) from plane 3 x+5 y+ 4 z=11 is 2 √2 units.2022 · asked 3×
- If a line makes an angle of (π)/4 with the positive directions of both x -axis and z -axis, then the angle…2024 · asked 3×
- Assertion: If a line makes angles α, β, γ with positive direction of the coordinate axes, then sin^2 α+ sin^2…2023 · asked 3×
- If the lines (x-3)/1=(1-y)/1=(z+2)/(p) and (2-x)/3=(y+1)/5=(z+56)/(2 p) are perpendicular to each other, then…2026 · asked 3×
- Find the equation of the plane passing through points (2, 1, 0), (3, -2, -2) and (1, 1, -7). Also, obtain its…2022 · asked 3×
- Direction ratios of a vector parallel to line x-1/2=-y=(2 z+1)/6 are:2024 · asked 3×
- Assertion ( A ): The lines r= a 1+λ b 1 and r= a 2+μ b 2 are perpendicular, when b 1 · b 2=0. Reason (R): The…2023 · asked 3×
CBSE Class 12 Mathematics past-paper question from the 2024board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.