MHT CET · Maths · Vector Algebra
A vector \(v\) is equally inclined to the \(x\) -axis, \(y\) -axis and \(z\) -axis respectively, its direction cosines are
- A \(\left. < \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right\rangle\)
- B \(\left. < -\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}\right\rangle\)
- C \( < \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}>\) or \( < -\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}>\)
- D None of the above
Answer & Solution
Correct Answer
(C) \( < \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}>\) or \( < -\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}>\)
Step-by-step Solution
Detailed explanation
Let the vector \(\mathbf{v}\) make an angle \(\alpha\) with each of the three axes, then direction cosine of \(\mathbf{v}\) are
\(
\begin{array}{l}
< \cos \alpha, \cos \alpha, \cos \alpha> \\
\text {Also, } \cos ^{2} \alpha+\cos ^{2} \alpha+\cos ^{2} \alpha=1 \\
\Rightarrow \cos ^{2} \alpha=1 / 3 \\
\Rightarrow \cos \alpha=\pm \frac{1}{\sqrt{3}}
\end{array}
\)
Hence, direction cosine of \(\mathbf{v}\) are
\( < \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}>
\)
Or
\( < -\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}>
\)
\(
\begin{array}{l}
< \cos \alpha, \cos \alpha, \cos \alpha> \\
\text {Also, } \cos ^{2} \alpha+\cos ^{2} \alpha+\cos ^{2} \alpha=1 \\
\Rightarrow \cos ^{2} \alpha=1 / 3 \\
\Rightarrow \cos \alpha=\pm \frac{1}{\sqrt{3}}
\end{array}
\)
Hence, direction cosine of \(\mathbf{v}\) are
\( < \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}>
\)
Or
\( < -\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}>
\)
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 \(P \equiv(-5,0), Q \equiv(0,0)\) and \(R \equiv(2,2 \sqrt{3})\) be three points. Then the equation of the bisector of the angle \(P Q R\) isMHT CET 2024 Medium
- The vector equation of the plane through the line of intersection of the planes \(x+y+z=1\) and \(2 x+3 y+4 z=5\), which is perpendicular to the plane \(x-y+z=0\), isMHT CET 2024 Easy
- The differential equation whose solution is \(y=c^2+\frac{c}{x}\), where \(c\) is constant, isMHT CET 2022 Easy
- If the vector \(\overline{\mathrm{c}}\) lies in the plane of \(\overline{\mathrm{a}}\) and \(\bar{b}\), where \(\bar{a}=\hat{i}-\hat{j}+2 \hat{k}, \bar{b}=\hat{i}+\hat{j}+\hat{k}\) and \(\overline{\mathrm{c}}=x \hat{\mathrm{i}}-(2-x) \hat{\mathrm{j}}-\hat{\mathrm{k}}\), then the value of \(x\) isMHT CET 2024 Easy
- If the vectors \(\overline{\mathrm{AB}}=3 \hat{\mathrm{i}}+4 \hat{\mathrm{k}}\) and \(\overline{\mathrm{AC}}=5 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}\) are the sides of the triangle \(A B C\), then the length of the median through A isMHT CET 2024 Medium
- If three distinct numbers are chosen randomly from first 100 natural numbers, then the probability that all three of them are divisible by both 2 and 3 isMHT CET 2022 Easy
More PYQs from MHT CET
- Given below are two statements.
Statement I: Hypothalamus is a link between nervous and endocrine system.
Statement II: Adenohypophysis secretes neurohormones.
In light of above statements, select the correct answer from the option given below.MHT CET 2022 Medium - The angular separation of the central maximum in the Fraunhofer diffraction pattern is measured. The slit is illuminated by the light of wavelength \(6000 Å\). If the slit is illuminated by light of another wavelength, the angular separation decreases by \(20 \%\). The wavelength of light used isMHT CET 2024 Easy
- The co-factors of the elements of second column of \(\left[\begin{array}{ccc}1 & -1 & 2 \\ 3 & 2 & 1 \\ -1 & 3 & 4\end{array}\right]\) areMHT CET 2021 Easy
- What is de Broglie's wavelength for a particle having mass \(6.64 \times 10^{-27} \mathrm{~kg}\) moving with velocity of \(3 \times 10^3 \mathrm{~ms}^{-1} ?\left[\mathrm{~h}=6.63 \times 10^{-34} \mathrm{Js}\right]\)MHT CET 2024 Easy
- The pressure and density of a diatomic gas \(\left(\gamma=\frac{7}{5}\right)\) changes adiabatically from \((\mathrm{P}, \rho)\) to \(\left(\mathrm{P}^{\prime}, \rho^{\prime}\right)\). If \(\frac{\rho^{\prime}}{\rho}=32\) then \(\frac{\mathrm{P}^{\prime}}{\mathrm{P}}\) should beMHT CET 2023 Hard
- Three critics review a book. For the three critics the odds in favor of the book are \(2: 5\), \(3: 4\) and \(4: 3\) respectively. The probability that the majority is in favor of the book, is given byMHT CET 2023 Hard