Vectors

Concept: The section formula gives the position vector of a point dividing a line segment internally in a given ratio. If a point divides the line joining points and internally in the ratio , then its position vector is a weighted average of the vectors of and . Let:

Position vector of
Position vector of
Position vector of
Suppose divides internally in the ratio , i.e.,
Step 1: Express vectors along the line Since lies on line , Given ratio: So in vector form:
Step 2: Solve algebraically Bring like terms together:
Step 3: Find position vector of Final Result (Section Formula): If a point divides a line segment internally in the ratio , then: Explanation: The formula shows that the point dividing the segment lies between the two endpoints and is a weighted average of their position vectors. The weights depend on the opposite segments of the ratio.

Concept: The area of a triangle formed by three points in 3D space is given by: where:



The magnitude of the cross product gives the area of the parallelogram.

Step 1: Find vectors
Step 2: Compute cross product Since the rows are proportional, the determinant is:
Step 3: Find area Final Answer: Explanation: The vectors and are parallel, meaning the three points are collinear. Hence, they do not form a triangle, and the area is zero.