JEE Mains · Maths · STD 12 - 11. three dimension geometry
Let a line \(L\) passing through the point \((1, 1, 1)\) be perpendicular to both the vectors \(2\hat{i} + 2\hat{j} + \hat{k}\) and \(\hat{i} + 2\hat{j} + 2\hat{k}\). If \(P(a, b, c)\) is the foot of perpendicular from the origin on the line \(L\), then the value of \(34(a + b + c)\) is :
- A \(50\)
- B \(80\)
- C \(100\)
- D \(120\)
Answer & Solution
Correct Answer
(C) \(100\)
Step-by-step Solution
Detailed explanation
The direction vector of the line \(L\) is given by the cross product of the two vectors to which it is perpendicular: \(\vec{d} = (2\hat{i} + 2\hat{j} + \hat{k}) \times (\hat{i} + 2\hat{j} + 2\hat{k}) = 2\hat{i} - 3\hat{j} + 2\hat{k}\) The equation of the line \(L\) passing…
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- Let \(\mathrm{f}:[0,3] \rightarrow \mathrm{R}\) be defined by \(f(x)=\min \{x-[x], 1+[x]-x\}\) where \([\mathrm{x}]\) is the greatest integer less than or equal to \(\mathrm{x}\). Let \(\mathrm{P}\) denote the set containing all \(x \in[0,3]\) where \(f\) is discontinuous, and \(Q\) denote the set containing all \(x \in(0,3)\) where \(f\) is not differentiable. Then the sum of number of elements in \(\mathrm{P}\) and \(\mathrm{Q}\) is equal to \(......\)JEE Mains 2021 Hard
- If the fractional part of the number \(\frac{{{2^{403}}}}{{15}}\) is \(\frac{k}{{15}}\), then \(k\) is equal toJEE Mains 2019 Hard
- Let \(a_1, a_2, a_3 \ldots a_n\) be \(n\) positive consecutive terms of an arithmetic progression. If \(d > 0\) is its common difference, then \(\lim _{n \rightarrow \infty} \sqrt{\frac{d}{n}}\left(\frac{1}{\sqrt{a_1}+\sqrt{a_2}}+\frac{1}{\sqrt{a_2}+\sqrt{a_3}}+\ldots \ldots .+\frac{1}{\sqrt{a_{n-1}}+\sqrt{a_n}}\right)\)JEE Mains 2023 Hard
- The sum of solutions of the equation \(\frac{\cos \mathrm{x}}{1+\sin \mathrm{x}}=|\tan 2 \mathrm{x}|, \mathrm{x} \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)-\left\{\frac{\pi}{4},-\frac{\pi}{4}\right\}\) is :JEE Mains 2021 Medium
- The number of points on the curve \(y=54 x^5-\) \(135 x^4-70 x^3+180 x^2+210 x\) at which the normal lines are parallel to \(x+90 y+2=0\) is :JEE Mains 2023 Hard
- Let \(A_1\) and \(A_2\) be two arithmetic means and \(G_1, G_2\), \(G _3\) be three geometric means of two distinct positive numbers. The \(G _1^4+ G _2^4+ G _3^4+ G _1^2 G _3^2\) is equal toJEE Mains 2023 Hard
More PYQs from JEE Mains
- The area (in \(sq. \,units\)) of the region, given by the set \(\left\{(x, y) \in R \times R \mid x \geq 0,2 x^{2} \leq y \leq 4-2 x\right\}\) is:JEE Mains 2021 Hard
- Let \(f ( x )=\left[2 x ^{2}+1\right]\) and \(g ( x )=\left\{\begin{array}{ll}2 x -3, & x < 0 \\ 2 x +3, & x \geq 0\end{array}\right.\), where \([t]\) is the greatest integer \(\leq t\) છે. Then, in the open interval \((-1,1)\), the number of points where fog is discontinuous is equal toJEE Mains 2022 Medium
- Which of the following statements is incorrect for the function \(g(\alpha)\) for \(\alpha \in R\) such that \(g(\alpha)=\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sin ^{\alpha} x}{\cos ^{\alpha} x+\sin ^{\alpha} x} d x\)JEE Mains 2021 Hard
- Let \( A(1,0) \), \( B(2,-1) \) and \( C(\frac{7}{3},\frac{4}{3}) \) be three points. If the equation of the bisector of the angle ABC is \( \alpha x+\beta y=5 \), then the value of \( \alpha^2+\beta^2 \) isJEE Mains 2026 Hard
- If \(g(x)=x^{2}+x-1\) and \((\operatorname{gof})(\mathrm{x})=4 \mathrm{x}^{2}-10 \mathrm{x}+5,\) then \(f\left(\frac{5}{4}\right)\) is equal toJEE Mains 2020 Hard
- The mirror image of the point \((1,2,3)\) in a plane is \(\left(-\frac{7}{3},-\frac{4}{3},-\frac{1}{3}\right) .\) Which of the following points lies on this plane ?JEE Mains 2020 Hard