Vectors

To solve the problem, we are given that a vector makes equal angles with all three coordinate axes and has a magnitude of units. We need to find the vector .

1. Direction Cosines of the Vector:
If a vector makes equal angles with the x-, y-, and z-axes, then its direction cosines are equal. Let the common direction cosine be .

Since the sum of the squares of direction cosines equals 1:

2. Unit Vector in Direction of :
The unit vector making equal angles with the axes is:

3. Multiply by Magnitude:
We now multiply the unit vector by the magnitude :

Final Answer:
The vector is .

To solve the problem, we are given two skew lines in symmetric form. We are to find the shortest distance between them.

1. Identify Direction Vectors and Points on Each Line:

First line:
Direction vector:
Point on the line: set the common ratio = 0 →
So, point

Second line:
Direction vector:
Point on the line: set the common ratio = 0 →
So, point

2. Vector Joining Points A and B:

3. Use the Formula for Shortest Distance Between Skew Lines:

4. Compute the Cross Product :

5. Compute Dot Product :

6. Magnitude of the Cross Product:

7. Final Calculation:

Final Answer:
The shortest distance between the lines is units.

The equation of a line in three-dimensional space parallel to a given vector and passing through a specific point can be determined using the vector equation of a line. The vector form of the line parallel to the vector and passing through the point is:

where , , and . Substituting these values into the vector equation, we get:

Breaking it into component form, the parametric equations are:

Therefore, the line can be represented parametrically as:

Step 1: Parametric Equations of the Lines

Line :

Points: , .

Direction vector of :

Parametric equations for line , using point and parameter :

Line :

Points: , .

Direction vector of :

Parametric equations for line , using point and parameter :

Step 2: Set Up Equations for Intersection

Equate the parametric equations to find a common point:

1. x-coordinate:
2. y-coordinate:
3. z-coordinate:

Step 3: Solve the Equations

From the z-coordinate:

Substitute into the x-coordinate:

Verify with the y-coordinate:

Left side: .

Right side: .

The equation holds, so , is consistent.

Step 4: Find the Intersection Point

For line , using :

Point: .

For line , using :

Point: .

The points match, confirming intersection at .

Step 5: Check for Skew or Coplanar Lines

Direction vectors: , .

Check if parallel by comparing ratios:

The lines are not parallel and share a point, so they intersect.

Final Answer

The lines intersect at:

(i) Complete the figure:

We are given:

From this, we can compute:

(ii) Find Vectors and

(iii-a) Given , , :

Step 1: Find angle between and

Step 2: Find

(iii-b) Given , :

Step 1: Find

Step 2: Find

Step 3: Cross Product

Step 4: Unit Vector

Final Answers:

3. Angle between and :
5. Unit vector =

If and are two mutually parallel vectors, then is equal to:

1. Understanding the Concepts:

If two vectors are parallel, then one is a scalar multiple of the other. That means:

for some scalar .

2. Writing the vectors in component form:

3. Setting up equations using the scalar multiple relationship:

From , we get:

• ❌

This contradicts the assumption. So, try equating component-wise ratios directly:

4. Check consistency:

These are not equal, so the vectors are not parallel with these components unless all three ratios are equal.

5. Set ratio from first two components:

Wait — the correct method is to match all ratios individually.

6. Use ratio of corresponding components:

Assuming vectors are parallel:

Final Answer:

The value of that makes the vectors parallel is 18.

We use the formula for the magnitude of the sum of vectors: Define: Step 1: Compute . Step 2: Compute . Step 3: Compute . Substituting values: Step 4: Take the square root. Thus, the final answer is:

The projection of a vector on a vector is given by: We are given that the projection is 4 units, so: First, calculate : Next, calculate : Now substitute into the projection formula: Solving for :

Given that and are unit vectors and the angle between them is , we are told that is also a unit vector.

We use the identity: Since and are unit vectors: So the expression becomes: Since is a unit vector, we set: Squaring both sides:

Answer:

Option (D)