Let \( \vec{a} = 2\hat{i} - 3\hat{j} + \hat{k} \), \( \vec{b} = 3\hat{i} + 2\hat{j} + 5\hat{k} \) and a vector \( \vec{c} \) be such that \[ (\vec{a} - \vec{c}) \times \vec{b} = -18\hat{i} - 3\hat{j} + 12\hat{k} \] and \[ \vec{a} \cdot \vec{c} = 3. \] If \( \vec{b} \times \vec{c} = \vec{d} \), then find \( |\vec{a} \cdot \vec{d}| \).
Let \( \vec{a} = 2\hat{i} - 3\hat{j} + \hat{k} \), \( \vec{b} = 3\hat{i} + 2\hat{j} + 5\hat{k} \) and a vector \( \vec{c} \) be such that \[ (\vec{a} - \vec{c}) \times \vec{b} = -18\hat{i} - 3\hat{j} + 12\hat{k} \] and \[ \vec{a} \cdot \vec{c} = 3. \] If \( \vec{b} \times \vec{c} = \vec{d} \), then find \( |\vec{a} \cdot \vec{d}| \).
Show Hint
- 18
- 12
- 9
- 15
The Correct Option is D
Approach Solution - 1
Approach Solution -2
We have vectors \(\mathbf{a}=\langle 2,-3,1\rangle\) and \(\mathbf{b}=\langle 3,2,5\rangle\). A vector \(\mathbf{c}\) satisfies \((\mathbf{a}-\mathbf{c})\times\mathbf{b}=\langle -18,-3,12\rangle\) and \(\mathbf{a}\cdot\mathbf{c}=3\). Define \(\mathbf{d}=\mathbf{b}\times\mathbf{c}\). We need \( |\mathbf{a}\cdot\mathbf{d}| \).
Concept Used:
Use the identity \((\mathbf{a}-\mathbf{c})\times\mathbf{b}=\mathbf{a}\times\mathbf{b}-\mathbf{c}\times\mathbf{b}=\mathbf{a}\times\mathbf{b}+\mathbf{b}\times\mathbf{c}=\mathbf{a}\times\mathbf{b}+\mathbf{d}\).
Step-by-Step Solution:
Step 1: Compute \(\mathbf{a}\times\mathbf{b}\).
\[ \mathbf{a}\times\mathbf{b}= \begin{vmatrix} \hat{i}&\hat{j}&\hat{k}\\ 2&-3&1\\ 3&2&5 \end{vmatrix} = (-17)\hat{i}+(-7)\hat{j}+13\hat{k}=\langle -17,-7,13\rangle. \]
Step 2: Use \((\mathbf{a}-\mathbf{c})\times\mathbf{b}=\mathbf{a}\times\mathbf{b}+\mathbf{d}\) to find \(\mathbf{d}\).
\[ \mathbf{d}=\langle -18,-3,12\rangle-\langle -17,-7,13\rangle=\langle -1,4,-1\rangle. \]
Step 3: Compute \(\mathbf{a}\cdot\mathbf{d}\).
\[ \mathbf{a}\cdot\mathbf{d}=\langle 2,-3,1\rangle\cdot\langle -1,4,-1\rangle =2(-1)+(-3)(4)+1(-1)=-2-12-1=-15. \]
Final Computation & Result
\[ |\mathbf{a}\cdot\mathbf{d}|=|-15|=15. \]
Answer: 15
Top JEE Main Mathematics Questions
- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
- If , then the general value of is:
- The general solution of
- If the constant term in the binomial expansion of $\left(\frac{x^{\frac{3}{2}}}{2}-\frac{4}{x^l}\right)^9$ is $-84$ and the coefficient of $x^{-3 l}$ is $2^\alpha \beta$, where $\beta<0$ is an odd number, then $|\alpha l-\beta|$ is equal to_____
Top JEE Main Vectors Questions
- Between the following two statements: {Statement-I:} Let \[ \vec{a} = \hat{i} + 2\hat{j} - 3\hat{k} \quad \text{and} \quad \vec{b} = 2\hat{i} + \hat{j} - \hat{k}. \] Then the vector \( \vec{r} \) satisfying \[ \vec{a} \times \vec{r} = \vec{a} \times \vec{b} \quad \text{and} \quad \vec{a} \cdot \vec{r} = 0 \] is of magnitude \( \sqrt{10} \). {Statement-II:} In a triangle \( \triangle ABC \), \[ \cos 2A + \cos 2B + \cos 2C \geq -\frac{3}{2}. \]
- Let \[ \vec{a} = 2\hat{i} + \alpha \hat{j} + \hat{k}, \quad \vec{b} = -\hat{i} + \hat{k}, \quad \vec{c} = \beta \hat{j} - \hat{k}, \] where \( \alpha \) and \( \beta \) are integers and \( \alpha \beta = -6 \). Let the values of the ordered pair \( (\alpha, \beta) \) for which the area of the parallelogram of diagonals \( \vec{a} + \vec{b} \) and \( \vec{b} + \vec{c} \) is \( \frac{\sqrt{21}}{2} \), be \( (\alpha_1, \beta_1) \) and \( (\alpha_2, \beta_2) \). Then \( \alpha_1^2 + \beta_1^2 - \alpha_2 \beta_2 \) is equal to:
- Let \(\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}\), \(\vec{b} = 2\hat{i} + 3\hat{j} - 5\hat{k}\), and \(\vec{c} = 3\hat{i} - \hat{j} + \lambda\hat{k}\) be three vectors. Let \(\vec{r}\) be a unit vector along \(\vec{b} + \vec{c}\). If \(\vec{r} \cdot \vec{a} = 3\), then \(3\lambda\) is equal to:
- Let \( L_1: \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4} \) and \( L_2: \frac{x-2}{3} = \frac{y-4}{4} = \frac{z-5}{5} \) be two lines. Then which of the following points lies on the line of the shortest distance between \( L_1 \) and \( L_2 \)?
- Let \( \vec{c} \) be the projection vector of \( \mathbf{b} = \lambda \hat{i} + 4 \hat{k}, \lambda > 0 \), on the vector \( \vec{a} = \hat{i} + 2 \hat{j} + 2 \hat{k} \). If \( |\vec{a} + \vec{c}| = 7 \), then the area of the parallelogram formed by the vectors \( \vec{b} \) and \( \vec{c} \) is:
Top JEE Main Questions
- In a hydrogen like atom, when an electron jumps from the $M$ - shell to the $L$ - shell, the wavelength of emitted radiation is $\lambda$. If an electron jumps from $N$-shell to the $L$-shell, the wavelength of emitted radiation will be :
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
What will be the equilibrium constant of the given reaction carried out in a \(5 \,L\) vessel and having equilibrium amounts of \(A_2\) and \(A\) as \(0.5\) mole and \(2 \times 10^{-6}\) mole respectively?
The reaction : \(A_2 \rightleftharpoons 2A\)- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.