VECTOR ALGEBRA | LINEAR COMBINATION LINEAR INDEPENDENCE AND LINEAR DEPENDENCE | Definition and physical interpretation: Linear Combination, Linear Combination: Linear Independence And Linear Dependence, Linearly Independent, Linearly Dependent, Theorem 1: If `veca` and `vecb` are two non collinear vectors; then every vector `vecr` coplanar with `veca` and `vecb` can be expressed in one and only one way as a linear combination: x`veca`+y`vecb`., Theorem 2: If `veca`, `vecb` and `vecc` are non coplanar vectors; then any vector `vecr` can be expressed as linear combination: x`veca`+y`vecb`+z`vecc`, Theorem 3:If vectors `veca`, `vecb` and `vecc` are coplanar then det(`veca` `vecb` `vecc`) = 0, Examples: Prove that the segment joining the middle points of two non parallel sides of a trapezium is parallel to the parallel sides and half of their sum., Components of a vector in terms of coordinates of its end points

Explanation

To prove that the segment joining the midpoints of two non-parallel sides of a trapezium is parallel to the parallel sides and half of their sum, we can follow these steps:

### Step 1: Define the trapezium and its points
Let the trapezium be \( ABCD \) where \( AB \) and \( CD \) are the parallel sides, and \( AD \) and \( BC \) are the non-parallel sides. Let \( M \) and \( N \) be the midpoints of sides \( AD \) and \( BC \) respectively.

### Step 2: Use the midpoint theorem
According to the midpoint theorem, the line segment joining the midpoints of two sides of a triangle is parallel to the third side and half as long. In our case, we will apply this theorem to triangles \( ABD \) and \( BCD \).

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