Vector Algebra

Concept:
When a point divides a line segment internally in a given ratio, its position vector can be found using the section formula. If point C lies on the line segment joining points A and B, and divides it internally in the ratio m:n, then the position vector of point C is given by:
C =

Given Equation:
2a + 3b − 5c = 0

Step 1: Rearranging the equation to isolate c:
5c = 2a + 3b
⇒ c =

Step 2: Now compare this with the internal division formula:
c =
We observe:
Numerator = 2a + 3b → implies A is associated with 2, B is associated with 3
Denominator = 5 = 2 + 3

So, the ratio in which point c divides the line segment AB internally is:
AB : 3:2 (from B to A)

Conclusion:
The point c divides the line joining points A and B internally in the ratio 3:2.
Hence, the correct option is (C) 3:2 internally.

We are given the vector equation: We will use the vector triple product identity to simplify this equation: Thus, the equation becomes: Now, since and are unit vectors, we focus on the coefficients of on both sides of the equation: This implies that the angle between and is . Next, we need to find the angle between and . From the equation, we can deduce that the angle between and is .
Thus, the correct answer is .

We are given the following vectors:

Step 1: Recall the formula for the dot product of two vectors:

The dot product of two vectors and is given by:

Step 2: Substitute the components of the vectors into the formula:

For and , we have:

Substitute these values into the dot product formula:

Conclusion:

The dot product is .