### Vector calculus in plane and space: Vectors in Planes and Space

### Linear combinations of vectors

*Scalar multiplications* and *additions* of vectors can be combined as follows.

Linear combinations of vectors

If #\vec{v}_1# , #\vec{v}_2, \ldots , \vec{v}_n# is a list of #n# vectors and #\lambda_1#, #\lambda_2, \ldots, \lambda_n# are real numbers, then

\[

\lambda_1 \cdot\vec{v}_1 + \lambda_2\cdot \vec{v}_2 + \cdots + \lambda_n\cdot \vec{v}_n

\]

is called a **linear combination** of the vectors #\vec{v}_1#, #\vec{v}_2, \ldots, \vec{v}_n#.

The term linear combination is used to give a name to vectors that we can construct from a given set of vectors using the two operations vector addition and scalar multiplication.

Calculating with vectors is relatively easy. Thanks to the associativity and distributivity, you can often skip a lot of formal steps when calculating.

We cannot simply multiply vectors. There is, however, an operation for vectors in the space, the *cross product*, which shares some of the properties related to the conventional multiplication of numbers.

This follows from the following calculation:

\[\begin{array}{rcl} 3\cdot\vec{u}-4\cdot\vec{v}&=&3\cdot\rv{9,-13}-4\cdot\rv{-6,-15}\\

&&\qquad \blue{\text{definitions }\vec{u}, \vec{v}}\\

&=&\rv{3\cdot 9, 3\cdot -13}+\rv{-4\cdot-6, -4\cdot -15}\\

&&\qquad \blue{\text{multiplication per coordinate}}\\

&=&\rv{27, -39}+\rv{24, 60}\\

&&\qquad \blue{\text{simplified}}\\

&=&\rv{27+24, -39+60}\\

&&\qquad \blue{\text{addition per coordinate}}\\

&=&\rv{51,21}\\

&&\qquad \blue{\text{simplified}}\end{array}\]

The figure below shows the vectors #\vec{u}# and #\vec{v}# drawn in black, the vectors #3\cdot \vec{u}# and #-4\cdot\vec{v}# in blue dashed lines, and the vector #3\cdot\vec{u}-4\cdot\vec{v}# in a red dotted line.

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