Explore the geometrical meaning of vector subtraction using this interactive tool. Vector subtraction A - B can be interpreted as A + (-B), where -B is the opposite vector of B.
If vectors \( \vec{A} \) and \( \vec{B} \) are given by their components as follows:
\[ \vec{A} = \langle u_1 , v_1 \rangle \quad \text{and} \quad \vec{B} = \langle u_2 , v_2 \rangle \]
then the components of vector \( \vec{A} - \vec{B} \) are given by:
\[ \vec{A} - \vec{B} = \langle u_1 - u_2 , v_1 - v_2 \rangle \]
Geometrically, \( \vec{A} - \vec{B} = \vec{A} + (-\vec{B}) \), where \(-\vec{B}\) is the vector with the same magnitude as \(\vec{B}\) but opposite direction.
Geometrical Interpretation: Vector subtraction A - B is equivalent to vector addition: A + (-B). This means we take vector B, reverse its direction to get -B, then add it to vector A. The resulting vector A - B goes from the tail of A to the head of -B when -B is placed at the head of A.
This approach makes it easier to understand subtraction as adding the opposite vector.