Let l1 and l2 be the lines r1 = λ(\(\hat{i}+\hat{j}+\hat{k}\)) and r2 = (\(\hat{j}-\hat{k}\)) + μ (\(\hat{i}+\hat{k}\)), respectively. Let X be the set of all the planes H containing line l1. For a plane H, let d (H) denote the smallest possible distance between the points of l2 and H. Let H0 be a plane in X for which d (H0) is the maximum value of d (H ) as H varies over all planes in X . Match each entry in List-I to the correct entries in List-II. List-I List-II (P) The value of d (H0) is (1) \(\sqrt3\) (Q) The distance of the point (0,1,2) from H0 is (2) \(\frac{1}{\sqrt3}\) (R) The distance of origin from H0 is (3) 0 (S) The distance of origin from the point of intersection of planes y = z, x = 1, and H0 is (4) \(\sqrt2\) (5) \(\frac{1}{\sqrt2}\)
The correct option is:
| List-I | List-II | ||
| (P) | The value of d (H0) is | (1) | \(\sqrt3\) |
| (Q) | The distance of the point (0,1,2) from H0 is | (2) | \(\frac{1}{\sqrt3}\) |
| (R) | The distance of origin from H0 is | (3) | 0 |
| (S) | The distance of origin from the point of intersection of planes y = z, x = 1, and H0 is | (4) | \(\sqrt2\) |
| (5) | \(\frac{1}{\sqrt2}\) | ||
- (P)\(\rightarrow\)(2) (Q)\(\rightarrow\)(4) (R)\(\rightarrow\)(5) (S)\(\rightarrow\)(1)
- (P)\(\rightarrow\)(5) (Q)\(\rightarrow\)(4) (R)\(\rightarrow\)(3) (S)\(\rightarrow\)(1)
- (P)\(\rightarrow \)(2) (Q)\(\rightarrow\)(1) (R)\(\rightarrow\)(3) (S)\(\rightarrow\)(2)
- (P)\(\rightarrow\)(5) (Q)\(\rightarrow\)(1) (R)\(\rightarrow\)(4) (S)\(\rightarrow\)(2)
The Correct Option is B
Solution and Explanation
Plane and Normal Vector Calculation
The correct option is (B).
Normal Vector of Plane Parallel \( l_1 \) and \( l_2 \):
The normal vector of the plane parallel to lines \( l_1 \) and \( l_2 \) is:
\[ \begin{vmatrix} i & j & k \\ 1 & 1 & 1 \\ 1 & 0 & 1 \end{vmatrix} = \hat{j}(1) - \hat{j}(1-1) + \hat{k}(-1) \]
\[ = \hat{i} - \hat{j} \]
Equation of Plane \( H_0 \):
\[ H_0 : x - z = c \quad \text{where} \quad (0, 0, 0) \]
\[ \Rightarrow c = 0 \]
\[ H_0 : x - z = 0 \]
Distance from the Point (0, 1, -1) to the Plane:
(P) The distance of point (0, 1, -1) from \( H_0 \) is:
\[ d(H_0) = \left| \frac{0 - (-1)}{\sqrt{2}} \right| = \frac{1}{\sqrt{2}} \quad \Rightarrow \quad P \rightarrow 5 \]
(Q) The distance from the point (0, 2, 0) to the plane is:
\[ d = \left| \frac{0 - 2}{\sqrt{2}} \right| = \sqrt{2} \quad \Rightarrow \quad Q \rightarrow 4 \]
(R) The distance from the origin to the plane is:
\[ d = \left| \frac{0}{\sqrt{2}} \right| = 0 \quad \Rightarrow \quad S \rightarrow 3 \]
Intersection Point and Distance:
(S) The point of intersection is (1, 1, 1):
\[ \Rightarrow S \rightarrow 1 \]
Distance is:
\[ d = \sqrt{1 + 1 + 1} = \sqrt{3} \]
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Vectors
The quantities having magnitude as well as direction are known as Vectors or Vector quantities. Vectors are the objects which are found in accumulated form in vector spaces accompanying two types of operations. These operations within the vector space include the addition of two vectors and multiplication of the vector with a scalar quantity. These operations can alter the proportions and order of the vector but the result still remains in the vector space. It is often recognized by symbols such as U ,V, and W
Representation of a Vector :
A line having an arrowhead is known as a directed line. A segment of the directed line has both direction and magnitude. This segment of the directed line is known as a vector. It is represented by a or commonly as AB. In this line segment AB, A is the starting point and B is the terminal point of the line.
Types of Vectors:
Here we will be discussing different types of vectors. There are commonly 10 different types of vectors frequently used in maths. The 10 types of vectors are:
- Zero vector
- Unit Vector
- Position Vector
- Co-initial Vector
- Like and Unlike Vectors
- Coplanar Vector
- Collinear Vector
- Equal Vector
- Displacement Vector
- Negative of a Vector