Vectors

We want to find the coefficient of in the expansion of

We can write

We are only interested in the coefficient of , so we can ignore terms of degree 2 or higher. Then

Therefore, the coefficient of is .

To solve this problem, we first perform the partial fraction decomposition of the given expression:

Step 1: Clear the denominator

Multiply both sides by to eliminate the denominators:

Step 2: Expand the right-hand side

Combine these terms:

Step 3: Equate coefficients

Compare the expanded form with the left-hand side .

Step 4: Solve for

Step 5: Compute the expression

Final Answer:

Step 1: Identify the Given Vectors Let the position vectors of the vertices be: We need to find the position vector of the point where the angle bisector of meets the line segment .
Step 2: Using the Angle Bisector Theorem By the angle bisector theorem, the point dividing in the ratio lies on the line joining and . Let be the point of intersection. By the angle bisector theorem: From the given position vectors: Since , the ratio is 1:1. Thus, the point is the midpoint of .
Step 3: Finding the Midpoint By the midpoint formula,
Step 4: Final Answer

We are given the four vectors , , , and . We are tasked with calculating .
Step 1: First, compute the cross product . Using the determinant formula for the cross product: Thus, .
Step 2: Now, compute the cross product : Thus, .
Step 3: Now, compute . We use the distributive property of the cross product: Thus, the result is .

We are given the following conditions: Using the property , we can write .
Now, to find the angle between and , we use the formula for the dot product: Substitute the known values and solve for , we find that: Thus, the correct answer is .

We are given the position vectors of points : To determine the type of triangle formed by points A, B, and C, we calculate the distances , , and using the distance formula between two points in 3D space: After computing the distances, we find that all the distances are equal, thus forming an equilateral triangle.
Thus, the correct answer is that A, B, and C form an equilateral triangle.

We are given: Step 1: Find the Vector Perpendicular to the Plane of and The vector perpendicular to both and is given by their cross product: Using the determinant form for the cross product, Step 2: Projection of on Let . The projection of on is given by: Calculating the dot product, Since the dot product is zero, the projection is zero. Step 3: Final Answer

Step 1: Formula for vector projection Step 2: Compute dot products Step 3: Compute projection

We want to find the coefficient of in the expansion of

We can write

We are only interested in the coefficient of , so we can ignore terms of degree 2 or higher. Then

Therefore, the coefficient of is .

To solve this problem, we first perform the partial fraction decomposition of the given expression:

Step 1: Clear the denominator

Multiply both sides by to eliminate the denominators:

Step 2: Expand the right-hand side

Combine these terms:

Step 3: Equate coefficients

Compare the expanded form with the left-hand side .

Step 4: Solve for

Step 5: Compute the expression

Final Answer:

To find the perimeter of triangle , we first determine the lengths of the sides , , and . The perimeter is the sum of these lengths. Step 1: Find the length of The vector is given by: The length of is: Step 2: Find the length of The vector is given by: The length of is: Step 3: Find the length of The vector is the negative of the sum of and : The length of is: Step 4: Compute the perimeter The perimeter of triangle is the sum of the lengths of its sides: Final Answer: