Vectors

Given: Points and

Step 1: Find the direction vector

The direction vector is given by:

Step 2: Write the vector equation of the line

The vector equation of the line passing through point with direction vector is:

Final Answer:

The given equations of the lines are:

Direction vectors:

Formula for angle between two lines:

Computing the dot product:

Computing magnitudes:

Substituting values:

Final Answer:

The given equation of the line is:

Direction vector of the line:

The given plane equation is:

Normal vector of the plane:

Condition for parallelism: The line is parallel to the plane if the direction vector is perpendicular to the normal vector , i.e.,

Computing the dot product:

Setting it to zero:

Solving for :

Final Answer:

Step 1: Use the dot product formula
The dot product of two vectors and is given by: Given: Also, we are given:

Step 2: Compute the dot product
Expanding using the formula:

Final Answer: is 4.

Step 1: Use the magnitude identity
The magnitude of the difference of two vectors is given by: Given: Squaring both sides:

Step 2: Solve for

Final Answer: is 9.

The equation of a plane is given by the formula: where is a point on the plane, and are the components of the normal vector to the plane.
Here, the normal vector is
So the components of the normal vector are , , and .
The given point is . Substituting these into the equation of the plane: Expanding this: Simplifying: Thus, the equation of the plane is:

The correct option is (D) :

We are given two vectors: We need to find a unit vector in the direction of .
Step 1: Find Simplify:
Step 2: Find the magnitude of The magnitude of is given by:
Step 3: Find the unit vector The unit vector in the direction of is given by: Substitute the values:

The correct option is (C) :

We are given:
- We need to calculate .
Step 1: Use the identity for the magnitude of the cross product The magnitude of the cross product is given by: So,
Step 2: Use the identity for the magnitude of the cross product We know that: Where is the angle between and . Thus, we have: ### Step 3: Use the dot product identity We are also given:
Step 4: Solve for Now we have the two equations: 1. 2. Squaring both equations: Adding these equations: Since , we have:

The correct option is (D) :

Given that , we can rewrite as .
The dot product properties and the given vectors' magnitudes can be used to find :
1. Substitute in the equation:
2. Expand using the properties of dot products:
3. Use magnitudes to solve for dot products:
- , thus . - , thus . - , thus . - implies , leading to .
4. Solve for :
5. Substitute back and solve:
6. Verification error, recalculation needed:
- Correct any missteps in derivation. The sum , confirming the correct answer from a detailed analysis based on given conditions and vector properties.

Step 1: We are given that .
Expanding this expression: Since and are unit vectors, and . Therefore: Simplifying: Step 2: Now, we need to calculate .
Using the distributive property of the dot product:
Simplifying each term: Using , , and , we get:
Simplifying further: