Show that each of the given three vectors is a unit vector: Which of the following is true?

Vector Algebra

1.
- A.
  
  4
- B.
  
  5
- C.
  
  6
- D.
  
  7
Correct Answer
A. 4
2.
- A.
  
  4
- B.
  
  -4
- C.
  
  5
- D.
  
  6
Correct Answer
B. -4
3.

Write the position vector of the mid-point of the vector joining the points P (2, 3, 4) and Q (4, 1, — 2).
- A.
- B.
- C.
- D.
Correct Answer
D.

Explanation
The position vector of the midpoint of the vector joining two points can be found by taking the average of the corresponding coordinates of the two points. In this case, the midpoint of the vector joining P(2, 3, 4) and Q(4, 1, -2) can be found by taking the average of the x-coordinates, y-coordinates, and z-coordinates. Therefore, the position vector of the midpoint is [(2+4)/2, (3+1)/2, (4+(-2))/2] = [3, 2, 1].

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2
0
4.
- A.
  
  0
- B.
  
  1
- C.
  
  2
- D.
  
  3
Correct Answer
A. 0
5.

Write unit vector in the direction of the sum of vectors:
- A.
- B.
- C.
- D.
Correct Answer
B.
6.

P and Q are two points with position vectors and respectively. Write the position vector of a point R which divides the line segment PQ in the ratio 2:1 externally.
- A.
- B.
- C.
- D.
Correct Answer
C.

Explanation
The position vector of point R can be found by using the external division formula. According to the formula, the position vector of R is given by (2Q + 1P) / (2+1). This means that we multiply the position vector of Q by 2, the position vector of P by 1, and then add them together. Finally, we divide the result by the sum of the ratios, which is 2+1 = 3.

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7.
- A.
  
  0
- B.
  
  1
- C.
  
  3
- D.
  
  4
Correct Answer
A. 0
8.

Using vectors, find the area of the triangle ABC with vertices A (1, 2, 3), B (2, -1, 4) and C (4, 5, -1).
- A.
- B.
- C.
- D.
Correct Answer
D.

Explanation
To find the area of a triangle using vectors, we can use the cross product of two vectors formed by the sides of the triangle. The magnitude of the cross product gives us the area of the parallelogram formed by the two vectors, and dividing it by 2 gives us the area of the triangle.

In this case, we can find the vectors AB and AC by subtracting the coordinates of the respective points. Then, we can calculate the cross product of AB and AC. Taking the magnitude of the cross product and dividing it by 2 gives us the area of triangle ABC.

Rate this question:
9.
- A.
  
  5
- B.
  
  -5
- C.
  
  Both A & B
- D.
  
  7
Correct Answer
C. Both A & B
10.
- A.
- B.
- C.
- D.
Correct Answer
A.
11.

Show that each of the given three vectors is a unit vector: Which of the following is true?
- A.
- B.
- C.
- D.
  
  All of these
Correct Answer
D. All of these

Explanation
The statement "All of these" means that each of the given three vectors is a unit vector. A unit vector is a vector with a magnitude of 1. Since the question states that all of the given vectors are unit vectors, it implies that each vector has a magnitude of 1, satisfying the definition of a unit vector. Therefore, the answer is correct.

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12.
- A.
- B.
- C.
- D.
Correct Answer
B.
13.
- A.
- B.
- C.
- D.
Correct Answer
A.
14.
- A.
  
  -1
- B.
  
  1
- C.
  
  -1.5
- D.
  
  2
Correct Answer
C. -1.5
15.
- A.
  
  Points are collinear
- B.
  
  Form a right angle triangle
- C.
  
  Doesn’t form a right triangle
- D.
  
  None of these
Correct Answer
B. Form a right angle triangle

Vector Algebra

Write the position vector of the mid-point of the vector joining the points P (2, 3, 4) and Q (4, 1, — 2).

Write unit vector in the direction of the sum of vectors:

P and Q are two points with position vectors and respectively. Write the position vector of a point R which divides the line segment PQ in the ratio 2:1 externally.

Using vectors, find the area of the triangle ABC with vertices A (1, 2, 3), B (2, -1, 4) and C (4, 5, -1).

Show that each of the given three vectors is a unit vector: Which of the following is true?