Question:medium
Which one of the following is a vector parallel to the straight line $\vec{r}=(\hat{i}-11\hat{j}+101\hat{k})+\lambda(3\hat{i}-5\hat{j}+2\hat{k}),\lambda\in\mathbb{R}$? ________.
Which one of the following is a vector parallel to the straight line $\vec{r}=(\hat{i}-11\hat{j}+101\hat{k})+\lambda(3\hat{i}-5\hat{j}+2\hat{k}),\lambda\in\mathbb{R}$? ________.
Show Hint
A vector $\vec{v}$ is parallel to $\vec{d}$ if $\vec{v} = k\vec{d}$.
Updated On: Jun 26, 2026
- $-3\hat{i}+5\hat{j}-2\hat{k}$
- $3\hat{i}+5\hat{j}+2\hat{k}$
- $\hat{i}-11\hat{j}+101\hat{k}$
- $-\hat{i}+11\hat{j}+101\hat{k}$
- $-4\hat{i}-16\hat{j}+103\hat{k}$
Show Solution
The Correct Option is A
Solution and Explanation
Step 1: Understanding the Concept
The equation of a line in vector form is \(\vec{r} = \vec{a} + \lambda\vec{d}\), where \(\vec{a}\) is the position vector of a point on the line, and \(\vec{d}\) is the direction vector of the line. The line is, by definition, parallel to its direction vector \(\vec{d}\). Any vector that is a non-zero scalar multiple of \(\vec{d}\) is also parallel to the line.
Step 2: Key Formula or Approach
1. Identify the direction vector \(\vec{d}\) from the given equation of the line.
2. Check which of the vectors in the options is a scalar multiple of \(\vec{d}\). That is, check if an option vector \(\vec{v}\) can be written as \(\vec{v} = k\vec{d}\) for some non-zero scalar \(k\).
Step 3: Detailed Explanation
1. Identify the direction vector.
The given equation of the line is:
\[ \vec{r} = (\hat{i} - 11\hat{j} + 101\hat{k}) + \lambda(3\hat{i} - 5\hat{j} + 2\hat{k}) \] By comparing this to the standard form \(\vec{r} = \vec{a} + \lambda\vec{d}\), we can identify the direction vector as:
\[ \vec{d} = 3\hat{i} - 5\hat{j} + 2\hat{k} \] 2. Check the options.
We are looking for a vector that is parallel to \(\vec{d}\). Let's examine each option.
(A) \(-3\hat{i} + 5\hat{j} - 2\hat{k}\):
Let's see if this vector is a multiple of \(\vec{d}\).
\(-3\hat{i} + 5\hat{j} - 2\hat{k} = -1 \times (3\hat{i} - 5\hat{j} + 2\hat{k})\).
This is equal to \(-1 \cdot \vec{d}\). Since it is a scalar multiple of \(\vec{d}\) (with \(k=-1\)), it is parallel to the line. This is the correct answer.
(B) \(3\hat{i} + 5\hat{j} + 2\hat{k}\):
The signs of the \(\hat{j}\) components are different. Not parallel.
(C) \(\hat{i} - 11\hat{j} + 101\hat{k}\):
This is the position vector \(\vec{a}\) of a point on the line. It is generally not parallel to the direction vector. Not parallel.
(D) \(-\hat{i} + 11\hat{j} + 101\hat{k}\):
Not a multiple of \(\vec{d}\).
(E) \(-4\hat{i} - 16\hat{j} + 103\hat{k}\):
Not a multiple of \(\vec{d}\).
Step 4: Final Answer
The vector \(-3\hat{i} + 5\hat{j} - 2\hat{k}\) is parallel to the given line.
The equation of a line in vector form is \(\vec{r} = \vec{a} + \lambda\vec{d}\), where \(\vec{a}\) is the position vector of a point on the line, and \(\vec{d}\) is the direction vector of the line. The line is, by definition, parallel to its direction vector \(\vec{d}\). Any vector that is a non-zero scalar multiple of \(\vec{d}\) is also parallel to the line.
Step 2: Key Formula or Approach
1. Identify the direction vector \(\vec{d}\) from the given equation of the line.
2. Check which of the vectors in the options is a scalar multiple of \(\vec{d}\). That is, check if an option vector \(\vec{v}\) can be written as \(\vec{v} = k\vec{d}\) for some non-zero scalar \(k\).
Step 3: Detailed Explanation
1. Identify the direction vector.
The given equation of the line is:
\[ \vec{r} = (\hat{i} - 11\hat{j} + 101\hat{k}) + \lambda(3\hat{i} - 5\hat{j} + 2\hat{k}) \] By comparing this to the standard form \(\vec{r} = \vec{a} + \lambda\vec{d}\), we can identify the direction vector as:
\[ \vec{d} = 3\hat{i} - 5\hat{j} + 2\hat{k} \] 2. Check the options.
We are looking for a vector that is parallel to \(\vec{d}\). Let's examine each option.
(A) \(-3\hat{i} + 5\hat{j} - 2\hat{k}\):
Let's see if this vector is a multiple of \(\vec{d}\).
\(-3\hat{i} + 5\hat{j} - 2\hat{k} = -1 \times (3\hat{i} - 5\hat{j} + 2\hat{k})\).
This is equal to \(-1 \cdot \vec{d}\). Since it is a scalar multiple of \(\vec{d}\) (with \(k=-1\)), it is parallel to the line. This is the correct answer.
(B) \(3\hat{i} + 5\hat{j} + 2\hat{k}\):
The signs of the \(\hat{j}\) components are different. Not parallel.
(C) \(\hat{i} - 11\hat{j} + 101\hat{k}\):
This is the position vector \(\vec{a}\) of a point on the line. It is generally not parallel to the direction vector. Not parallel.
(D) \(-\hat{i} + 11\hat{j} + 101\hat{k}\):
Not a multiple of \(\vec{d}\).
(E) \(-4\hat{i} - 16\hat{j} + 103\hat{k}\):
Not a multiple of \(\vec{d}\).
Step 4: Final Answer
The vector \(-3\hat{i} + 5\hat{j} - 2\hat{k}\) is parallel to the given line.
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