Three Dimensional Geometry

The direction cosines of a line in three-dimensional space are the cosines of the angles that the line makes with the positive x, y, and z axes. For the z-axis itself, these cosines can be determined as follows:

Therefore, the direction cosines of the z-axis are 0,0,1, which corresponds to the option .

Step 1: Understanding the Concept:
The vector equation of a line is given in the form , where is the position vector of a point on the line and is a vector parallel to the line, whose components give the direction ratios.
Step 3: Detailed Explanation:
The given equation is .
By comparing with , we have:


(A) A point on the given line:
The position vector corresponds to a point on the line. The coordinates of this point are (1, -2, 4). This matches with (III).
(B) Direction ratios of the line:
The components of the parallel vector are the direction ratios of the line. These are (-1, 2, -4). This matches with (IV).
(C) Direction cosines of the line:
Direction cosines are the components of the unit vector in the direction of . First, find the magnitude of .
.
The unit vector is .
The direction cosines are . This matches with (I).
(D) Direction ratios of a line perpendicular to given line:
Let the direction ratios of a perpendicular line be . The dot product of its direction vector with must be zero.

.
We need to check which direction ratios from List-II satisfy this condition. Let's test option (II) (4, -2, -2).
.
The condition is satisfied. So, (4, -2, -2) are direction ratios of a perpendicular line. This matches with (II).
Step 4: Final Answer:
The correct matching is (A) - (III), (B) - (IV), (C) - (I), (D) - (II). This corresponds to option (3).

Step 1: Understanding the Concept:
We are asked to find the shortest distance between two lines given in Cartesian form. First, we need to determine if the lines are parallel, intersecting, or skew.
Step 2: Key Formula or Approach:
The Cartesian form of a line is . The vector form is . The direction ratios of the first line are and for the second line are . Two lines are parallel if their direction ratios are proportional, i.e., . If lines are parallel, the shortest distance is the perpendicular distance from a point on one line to the other line. If the lines are not parallel, they are either intersecting or skew.
Step 3: Detailed Explanation:
Let the first line be L1 and the second line be L2.
L1: . The direction ratios are . L2: . The direction ratios are .
First, check if the lines are parallel by comparing their direction ratios.
Since the ratios are equal, the direction vectors are proportional ( ). Therefore, the lines are parallel.
Now, to find the distance between them, we need to check if they are the same line (coincident) or distinct parallel lines. We can do this by taking a point from one line and checking if it lies on the other.
A point on L1 is . Let's check if this point satisfies the equation of L2.
Substitute the coordinates of into the equation for L2:
Since , the point (1, 2, 3) does not lie on the second line. Therefore, the lines are distinct and parallel.
The shortest distance between two parallel lines and is given by the formula:
From the equations:


(we can use the simpler direction vector)

Now calculate the cross product:
Magnitude of the cross product: .
Magnitude of the direction vector: .
Distance: .

Step 1: Understanding the Concept:
The angles are the direction angles of a line. The cosines of these angles, , are called the direction cosines of the line. There is a fundamental identity relating these direction cosines.
Step 2: Key Formula or Approach:
The fundamental identity for direction cosines ( ) is:
where , , and .
So, .
We also use the Pythagorean identity from trigonometry: .
Step 3: Detailed Explanation:
We need to find the value of the expression .
Using the identity , we can rewrite the expression:
Rearranging the terms:
Now, substitute the fundamental identity for direction cosines, :
Step 4: Final Answer:
The value of is 2.

Step 1: Understanding the Concept:
We are asked to find the shortest distance between two lines given in Cartesian form. First, we need to determine if the lines are parallel, intersecting, or skew.
Step 2: Key Formula or Approach:
The Cartesian form of a line is . The vector form is . The direction ratios of the first line are and for the second line are . Two lines are parallel if their direction ratios are proportional, i.e., . If lines are parallel, the shortest distance is the perpendicular distance from a point on one line to the other line. If the lines are not parallel, they are either intersecting or skew.
Step 3: Detailed Explanation:
Let the first line be L1 and the second line be L2.
L1: . The direction ratios are . L2: . The direction ratios are .
First, check if the lines are parallel by comparing their direction ratios.
Since the ratios are equal, the direction vectors are proportional ( ). Therefore, the lines are parallel.
Now, to find the distance between them, we need to check if they are the same line (coincident) or distinct parallel lines. We can do this by taking a point from one line and checking if it lies on the other.
A point on L1 is . Let's check if this point satisfies the equation of L2.
Substitute the coordinates of into the equation for L2:
Since , the point (1, 2, 3) does not lie on the second line. Therefore, the lines are distinct and parallel.
The shortest distance between two parallel lines and is given by the formula:
From the equations:


(we can use the simpler direction vector)

Now calculate the cross product:
Magnitude of the cross product: .
Magnitude of the direction vector: .
Distance: .

Step 1: Understanding the Concept:
The angles are the direction angles of a line. The cosines of these angles, , are called the direction cosines of the line. There is a fundamental identity relating these direction cosines.
Step 2: Key Formula or Approach:
The fundamental identity for direction cosines ( ) is:
where , , and .
So, .
We also use the Pythagorean identity from trigonometry: .
Step 3: Detailed Explanation:
We need to find the value of the expression .
Using the identity , we can rewrite the expression:
Rearranging the terms:
Now, substitute the fundamental identity for direction cosines, :
Step 4: Final Answer:
The value of is 2.

Step 1: Understanding the Concept:
The vector equation of a line is given in the form , where is the position vector of a point on the line and is a vector parallel to the line, whose components give the direction ratios.
Step 3: Detailed Explanation:
The given equation is .
By comparing with , we have:


(A) A point on the given line:
The position vector corresponds to a point on the line. The coordinates of this point are (1, -2, 4). This matches with (III).
(B) Direction ratios of the line:
The components of the parallel vector are the direction ratios of the line. These are (-1, 2, -4). This matches with (IV).
(C) Direction cosines of the line:
Direction cosines are the components of the unit vector in the direction of . First, find the magnitude of .
.
The unit vector is .
The direction cosines are . This matches with (I).
(D) Direction ratios of a line perpendicular to given line:
Let the direction ratios of a perpendicular line be . The dot product of its direction vector with must be zero.

.
We need to check which direction ratios from List-II satisfy this condition. Let's test option (II) (4, -2, -2).
.
The condition is satisfied. So, (4, -2, -2) are direction ratios of a perpendicular line. This matches with (II).
Step 4: Final Answer:
The correct matching is (A) - (III), (B) - (IV), (C) - (I), (D) - (II). This corresponds to option (3).