Vectors

For three vectors to be coplanar, their scalar triple product must be zero. That is, the determinant of the matrix formed by the components of the vectors must be zero.

Expanding the determinant:

Therefore, the value of is 6.

Thus, the correct option is (A) 6.

Given:
Let be the angle between and . Then:

Step 1: Use identities
Plugging into the given equation: Let . Since : Divide both sides by 3: Step 2: Try values from options
Try :
Equation becomes: Since : Both values are possible ⇒ equation holds for

✅ Correct answer: 9

The correct answer is: (C):

We are tasked with finding the component of in the direction of the vector .

Step 1: Formula for the component of a vector

The component of a vector in the direction of a vector is given by the formula:

Step 2: Compute the dot product

Let and . The dot product is:

Step 3: Find the magnitude of

The magnitude of vector is:

Step 4: Compute the component

Now we can compute the component of in the direction of using the formula:

Conclusion:
The component of in the direction of is , so the correct answer is (C): .

1. Understand the condition:

Given two unit vectors and with angle between them, we need to find when is also a unit vector.

2. Compute the magnitude of :

For to be a unit vector:

3. Solve for :

Correct Answer: (C)

1. Find the midpoint vector:

Given vectors and .

The median through is the vector from to the midpoint of :

2. Compute the length of the median:

Correct Answer: (C)