JEE Mains · Maths · STD 12 - 10. vector algebra
Let \(\vec{a}=2 \hat{i}+3 \hat{j}+4 \hat{k}, \vec{b}=2 \hat{i}-2 \hat{j}-2 \hat{k}\) and \(\overrightarrow{ c }=-\hat{ i }+4 \hat{ j }+3 \hat{ k }\). If \(\overrightarrow{ d }\) is a vector perpendicular to both \(\vec{b}\) and \(\overrightarrow{ c }\) and \(\overrightarrow{ a } \cdot \overrightarrow{ d }=18\), Then \(|\overrightarrow{ a } \times \overrightarrow{ d }|^2\) is equal to \(..........\).
- A \(640\)
- B \(760\)
- C \(680\)
- D \(720\)
Answer & Solution
Correct Answer
(D) \(720\)
Step-by-step Solution
Detailed explanation
\(\overrightarrow{ a }=\lambda(\overrightarrow{ b } \times \overrightarrow{ c })\) \(\overrightarrow{ b } \times \overrightarrow{ c }=\left|\begin{array}{ccc}\hat{ i } & \hat{ j } & \hat{ k } \\ 1 & -2 & -2 \\ -1 & 4 & 3\end{array}\right|=2 \hat{i}-\hat{ j }+2 \hat{ k }\)…
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
- If the fourth term in the binomial expansion of \(\left(\sqrt{\frac{1}{x^{1+\log _{10} x}}}+x^{\frac{1}{12}}\right)^{6}\) is equal to \(200\), and \(x > 1\), then the value of \(x\) isJEE Mains 2019 Hard
- A man is observing, from the top of a tower, a boat speeding towards the tower from a certain point A, with uniform speed. At that point, angle of depression of the boat with the man's eye is \(30^{\circ}\) (Ignore man's height). After sailing for \(20\) seconds, towards the base of the tower (which is at the level of water), the boat has reached a point \(B\), where the angle of depression is \(45^{\circ}\). Then the time taken (in seconds) by the boat from \(B\) to reach the base of the tower isJEE Mains 2021 Hard
- The difference between degree and order of a differential equation that represents the family of curves given by \(y^{2}=a\left(x+\frac{\sqrt{a}}{2}\right), a>0\) isJEE Mains 2021 Medium
- Let \(P\) the point of intersection of the lines \(\frac{x-2}{1}=\frac{y-4}{5}=\frac{z-2}{1}\) and \(\frac{x-3}{2}=\frac{y-2}{3}=\frac{z-3}{2}\). Then, the shortest distance of \(\mathrm{P}\) from the line \(4 \mathrm{x}=2 \mathrm{y}=\mathrm{z}\) isJEE Mains 2024 Medium
- Let \(C_1\) be the circle in the third quadrant of radius 3 , that touches both coordinate axes. Let \(\mathrm{C}_2\) be the circle with centre \((1,3)\) that touches \(\mathrm{C}_1\) externally at the point \((\alpha, \beta)\). If \((\beta-\alpha)^2=\frac{m}{n}, \operatorname{gcd}(m, n)=1\), then \(m+n\) is equal to :JEE Mains 2025 Medium
- Let a curve \(y=f(x), x \in(0, \infty)\) pass through the points \(P\left(1, \frac{3}{2}\right)\) and \(Q\left(a, \frac{1}{2}\right)\). If the tangent at any point \(R(b, f(b))\) to the given curve cuts the \(y\)-axis at the point \(S(0, c)\) such that \(b c=3\), then \((P Q)^2\) is equal to \(.........\).JEE Mains 2023 Hard
More PYQs from JEE Mains
- The smallest natural number \(n,\) such that the coefficient of \(x\) in the expansion of \({\left( {{x^2}\, + \,\frac{1}{{{x^3}}}} \right)^n}\) is \(^n{C_{23}}\) isJEE Mains 2019 Hard
- \(\mathop {\lim }\limits_{n \to \infty } {\left( {\frac{{\left( {n + 1} \right)\left( {n + 2} \right) \ldots .\;3n}}{{{n^{2n}}}}} \right)^{\frac{1}{n}}} = \)JEE Mains 2016 Hard
- The square of the distance of the point \((-2, -8, 6)\) from the line \(\dfrac{x-1}{1} = \dfrac{y-1}{2} = \dfrac{z}{-1}\) along the line \(\dfrac{x+5}{1} = \dfrac{y+5}{-1} = \dfrac{z}{2}\) is equal to:JEE Mains 2026 Hard
- Let \(A = \begin{bmatrix} 1 & 3 & -1 \\ 2 & 1 & \alpha \\ 0 & 1 & -1 \end{bmatrix}\) be a singular matrix. Let \(f(x) = \int\limits_0^x (t^2 + 2t + 3)\,dt\), \(x \in [1, \alpha]\). If \(M\) and \(m\) are respectively the maximum and the minimum values of \(f\) in \([1, \alpha]\), then \(3(M - m)\) is equal to :JEE Mains 2026 Hard
- The plane \(2 x-y+z=4\) intersects the line segment joining the points \(A ( a ,-2\), 4) and \(B (2, b ,-3)\) at the point \(C\) in the ratio \(2: 1\) and the distance of the point \(C\) from the origin is \(\sqrt{5}\). If \(ab <0\) and \(P\) is the point \(( a - b , b , 2 b - a )\) then \(CP ^2\) is equal to :JEE Mains 2023 Hard
- Let \( f :R\rightarrow R \) be a twice differentiable function such that \( f^{\prime\prime}(x)>0 \) for all \( x\in R \) and \( f^{\prime}(a-1)=0 \), where a is real number. Let \( g(x)=f(tan^{2}x-2~tan~x+a), 0 < x < \frac{\pi}{2}\).
Consider the following two statements :
(I) g is increasing in \( (0,\frac{\pi}{4}) \)
(II) g is decreasing in \( (\frac{\pi}{4},\frac{\pi}{2}) \)
Then,JEE Mains 2026 Easy