MHT CET · Maths · Vector Algebra
The position vectors of vertices of \(\triangle A B C\) are \(4 \hat{i}-2 \hat{j}\); \(\hat{i}+4 \hat{j}-3 \hat{k}\) and \(-\hat{i}+5 \hat{j}+\hat{k}\) respectively, then \(m \angle A B C=\)
- A \(\frac{\pi}{3}\)
- B \(\frac{\pi}{4}\)
- C \(\frac{\pi}{6}\)
- D \(\frac{\pi}{2}\)
Answer & Solution
Correct Answer
(D) \(\frac{\pi}{2}\)
Step-by-step Solution
Detailed explanation
\(\angle A B C=\text { angle between } \overrightarrow{B A} \text { and } \overrightarrow{B C} \)
\( =\cos ^{-1}\left(\frac{\overrightarrow{B A} \cdot \overrightarrow{B C}}{|\overrightarrow{B A}||\overrightarrow{B C}|}\right)=\cos ^{-1}\)\(\left(\frac{(3 \hat{i}-6 \hat{j}+3 \hat{k}) \cdot(-2 \hat{i}+\hat{j}+4 \hat{k})}{\sqrt{3^2+(-6)^2+3^2} \sqrt{(-2)^2+1^2+4^2}}\right) \)
\( =\cos ^{-1}\left(\frac{6-6+12}{\sqrt{54} \sqrt{21}}\right)=\cos ^{-1}(0)=\frac{\pi}{2}\)
\( =\cos ^{-1}\left(\frac{\overrightarrow{B A} \cdot \overrightarrow{B C}}{|\overrightarrow{B A}||\overrightarrow{B C}|}\right)=\cos ^{-1}\)\(\left(\frac{(3 \hat{i}-6 \hat{j}+3 \hat{k}) \cdot(-2 \hat{i}+\hat{j}+4 \hat{k})}{\sqrt{3^2+(-6)^2+3^2} \sqrt{(-2)^2+1^2+4^2}}\right) \)
\( =\cos ^{-1}\left(\frac{6-6+12}{\sqrt{54} \sqrt{21}}\right)=\cos ^{-1}(0)=\frac{\pi}{2}\)
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
- The equation of the line passing through the point \((2,3,-4)\) and perpendicular to
\(\mathrm{XOZ}\) plane isMHT CET 2020 Medium - Negation of the statement
"The payment will be made if and only if the work is finished in time." IsMHT CET 2023 Easy - The lengths of sides of a triangle are three consecutive natural numbers and its largest angle is twice the smallest one. Then the length of the sides of the triangle (in units) areMHT CET 2023 Easy
- If \(E_{1}\) denotes the events of coming sum 6 in throwing two dice and \(E_{2}\) be the event of coming 2 in any one of the two, then \(P\left(E_{2} / \mathrm{E}_{1}\right.\) ) isMHT CET 2011 Medium
- Consider three observations \(a, b\) and \(c\) such that \(b=a+c\). If the standard deviation of \(\mathrm{a}+2, \mathrm{~b}+2, \mathrm{c}+2\) is d, then what holds true.MHT CET 2024 Easy
- If \(\tan ^{-1}\left(\frac{1-x}{1+x}\right)=\frac{1}{2} \tan ^{-1} x\), then \(x\) isMHT CET 2023 Easy
More PYQs from MHT CET
- Which from following sentences is NOT correct?MHT CET 2023 Easy
- When light of wavelength \(\lambda\) is incident on photosensitive surface, the stopping potential is \(V\). When light of wavelength \(3 \lambda\) is incident on same surface, the stopping potential is \(\frac{V}{6}\). Threshold wavelength for the surface isMHT CET 2016 Medium
- Given below are two statements.
Statement I - Smaller the food chain, more is the available energy at each trophic level.
Statement II - Pyramid of energy can also be sometimes inverted.
In the light of above statements, Select the correct option given below:MHT CET 2025 Hard - A particle of mass ' \(m\) ' performs uniform circular motion of radius ' \(r\) ' with linear speed ' \(v\) ' under the application of force ' \(F\) '. If ' \(m\) ', ' \(v\) ' and ' \(r\) ' are all increased by \(20 \%\) the necessary change in force required to maintain the particle in uniform circular motion, isMHT CET 2024 Medium
- Which among the following is NOT a feature of \(\mathrm{S}_{\mathrm{N}} 2\) mechanism?MHT CET 2024 Medium
- An organic compound was found to contain \(40.0 \% \mathrm{C}\) and \(6.66 \% \mathrm{H}\). Find it's molecular formula (molar mass \(=180\) )MHT CET 2020 Easy