How to Solve Vector Equations

To solve vector equations, for each vector, gather the factors that are in front of it. Do this with all the vectors in the equation to make a system of equations.

Theory

Vector Equations

A vector equation is expressed as

a \vec{u} + b \vec{v} = c \vec{u} + d \vec{v},

where $\vec{u}$ and $\vec{v}$ are two non-parallel vectors, and $a$ , $b$ , $c$ and $d$ are expressions that can include both constants and variables.

Set the expressions in front of $\vec{u}$ equal to each other and the expressions in front of $\vec{v}$ equal to each other:

a = c b = d .

This is your system of equations.

Example 1

Find the values of $k$ and $l$ given that

(2 - k) \vec{u} + \vec{v} = 4 \vec{u} - (l + 3) \vec{v} .

You put the expressions in front of $\vec{u}$ equal to each other and the expressions in front of $\vec{v}$ equal to each other. Then you can solve the set of equations:

\begin{array}{l} 2 - k & = 4 & 1 & = - l - 3 \\ k & = - 2 & l & = - 4 \end{array}

Example 2

Find the values of $k$ and $l$ given that

(l + 3 k) \vec{u} + 3 \vec{v} = 2 k \vec{u} - (4 + 2 k) \vec{v} .

You put the expressions in front of $\vec{u}$ equal to each other and the expressions in front of $\vec{v}$ equal to each other. That gives you this system of equations, which you can solve:

\begin{array}{l} l + 3 k & = 2 k & 3 & = - 4 - 2 k \\ 7 & = - 2 k \\ k & = - \frac{7}{2} \\ l & = - k \\ l & = - (- \frac{7}{2}) \\ l & = \frac{7}{2} \end{array}

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