Scaling Vectors

"Scaling" \(x\) by a number \(\lambda\) involves multiplying each element in \(x\) by \(\lambda\). Visually (for perpendicular axes) this expands or shrinks \(x\) without changing the orientation of the vector. When \(\lambda\) is negative, the direction of the vector flips and points in the opposite direction. For parallel axes, this increases, shrinks, and/or flips the heights of each element by a the same value (\(\lambda\)).

Adding Vectors

Adding vectors is done elementwise. Visually (for perpendicular axes), this sum is a new vector given by placing the two vectors "tip-to-tail." For parallel axes, we simply "stack" the heights together to get the height of the new vector.

Adding multiple vectors is illustrated below in 2D and in 3D

UPDATE 6: VISUALIZATION OF ADDING VECTORS IS SHOWN HERE.

Vector Addition in 2D

Vector Addition in 3D

Subtracting Vectors

Subtraction between two vectors is also done element-wise. If we think of the vector as floating rather than attached to the origin, the difference vector points from the end of one vector to the other.

Vector Difference in 2D

Vector Difference in 3D