Linear Combinations
A vector is said to be a Linear Combination of another vector if there exists a non-zero scalar such that \( \vec{u} = k\vec{v} \) or \( \vec{u} = a\vec{v} + b\vec{w} \) etc. In general:
\( \vec{u} = \sum_{i=0}^n a_i\vec{v}_i = a_1\vec{v}_1 + a_2\vec{v}_2 + \cdots + a_n\vec{v}_n \)
Where \( a_1, a_2, \) etc. are scalars and \( \vec{v}_1, \vec{v}_2, \) etc. are vectors.
If \( \vec{u} = k\vec{v} \), \( \vec{u} \) is said to be Collinear with respect to \( \vec{v} \). In other words, \( \vec{u} \) is a scalar multiple of \( \vec{v} \), \( \vec{u} \) is parallel to \( \vec{v} \) and \( \vec{u} \) and \( \vec{v} \) point in the same direction.
\(\vec{u} \parallel \vec{v} \)
\(\vec{u} = 2.5 \vec{v} \)
If \( \vec{u} = a\vec{v} + b\vec{w} \), \( \vec{u} \) is said to be Coplanar with respect to \( \vec{v} \) and \( \vec{w} \). In other words, \( \vec{u} \), \( \vec{v} \) and \( \vec{w} \) all exist on the same plane in 3-D space.
\(\vec{w} = 2.5 \vec{v} + 2 \vec{u} \)
In 3-D, imagine that all three vectors can be drawn on a piece of paper.
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Standard Basis Form
Recall the Standard Basis unit vectors that are in the direction of the the x, y and z axes:
\( \hat{\imath} = \left<1,0,0\right> \)
\( \hat{\jmath} = \left<0,1,0\right> \)
\( \hat{k} = \left<0,0,1\right> \)
We can express any vector in 3-D as a linear combination of these vectors. This is called Standard Basis Form