CBSE 2025 · Region 7 · Set 1 · Q31 · 3 marks
The scalar product of the vector $\displaystyle \vec{a}=\hat{i}-\hat{j}+2 \hat{k}$ with a unit vector along sum of vectors $\displaystyle \vec{b}=2 \hat{i}-4 \hat{j}+5 \hat{k}$ and $\displaystyle \vec{c}=\lambda \hat{i}-2 \hat{j}-3 \hat{k}$ is equal to $\displaystyle 1$ . Find the value of $\displaystyle \lambda$.Find the shortest distance between the lines: \[\begin{aligned} & \vec{r}=(2 \hat{i}-\hat{j}+3 \hat{k})+\lambda(\hat{i}-2 \hat{j}+3 \hat{k}) \\ & \vec{r}=(\hat{i}+4 \hat{k})+\mu(3 \hat{i}-6 \hat{j}+9 \hat{k}) \end{aligned} \]
The scalar product of the vector $\displaystyle \vec{a}=\hat{i}-\hat{j}+2 \hat{k}$ with a unit vector along sum of vectors $\displaystyle \vec{b}=2 \hat{i}-4 \hat{j}+5 \hat{k}$ and $\displaystyle \vec{c}=\lambda \hat{i}-2 \hat{j}-3 \hat{k}$ is equal to $\displaystyle 1$ . Find the value of $\displaystyle \lambda$.
Find the shortest distance between the lines: \[\begin{aligned} & \vec{r}=(2 \hat{i}-\hat{j}+3 \hat{k})+\lambda(\hat{i}-2 \hat{j}+3 \hat{k}) \\ & \vec{r}=(\hat{i}+4 \hat{k})+\mu(3 \hat{i}-6 \hat{j}+9 \hat{k}) \end{aligned} \]
Marking-scheme solution
(a)
Let $\displaystyle \overrightarrow{\mathbf{d}}=\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}=(\mathbf{2}+\boldsymbol{\lambda}) \hat{\mathbf{i}}-\mathbf{6} \hat{\mathbf{j}}+\mathbf{2} \hat{\mathbf{k}}$
\[\hat{\mathbf{d}}=\frac{(2+\lambda) \hat{\mathbf{i}}-6 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}}{\sqrt{(2+\lambda)^{2}+40}}
\]
Vector AlgebraProduct of Two VectorsApplyshort_answermedium
More from Vector Algebra
- If a+ b+ c= 0 such that a =3, b =5, c =7, then find the angle between a and b. OR If a and b are unit vectors…2025 · asked 3×
- The position vectors of points P and Q are p and q respectively. The point R divides line segment PQ in the…2024 · asked 3×
- If a+ b= i and a=2 i-2 j+2 k, then b equals:2023 · asked 3×
- A vector a makes equal angles with all the three axes. If the magnitude of the vector is 5 √3 units, then…2025 · asked 3×
- The vector with terminal point A(2,-3,5) and initial point B(3,-4,7) is:2024 · asked 3×
- Find the position vector of point C which divides the line segment joining points A and B having position…2024 · asked 3×
- If a =2 and -3 ≤ k ≤ 2, then k a ∈:2024 · asked 3×
- Unit vector along PQ, where coordinates of P and Q respectively are (2,1,-1) and (4,4,-7), is2023 · asked 3×
CBSE Class 12 Mathematics past-paper question from the 2025board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.