Question

Consider the vectors$\overrightarrow{\mathbf{u}}\mathbf{,}\overrightarrow{\mathbf{v}}$and$\overrightarrow{\mathbf{w}}$sketched in the accompanying figure. Find the coordinate vector of $\overrightarrow{\mathbf{w}}$with respect to the basis$\overrightarrow{\mathbf{u}}\mathbf{,}\overrightarrow{\mathbf{v}}$.

Short answer

The coordinate vector of $\overrightarrow{w}$with respect to the basis$\overrightarrow{u},\overrightarrow{v}$ is-

${\left[\overrightarrow{w}\right]}_{؏}=\left[\begin{array}{c}-1\\ -1\end{array}\right],$where$؏=\left\{\overrightarrow{u},\overrightarrow{v}\right\}$

Step-by-step solution

 Finding the vector w→

Let$؏=\left\{\overrightarrow{u},\overrightarrow{v}\right\}$ be the basis.

We know that the if resultant of two vector $\overrightarrow{u},\overrightarrow{v}$is $\overrightarrow{w}$.

Then$\overrightarrow{w}=\overrightarrow{u}+\overrightarrow{v}$

But from the figure we can see that the vector$\overrightarrow{w}$ is in the opposite direction of$\overrightarrow{\mathrm{u}}\mathrm{and}\overrightarrow{\mathrm{v}}$, therefore we have

$$\begin{gathered} -\overrightarrow{w}=\overrightarrow{u}+\overrightarrow{v} \\ \Rightarrow \overrightarrow{w}=-\overrightarrow{u}-\overrightarrow{v} \\ \Rightarrow \overrightarrow{w}=(-1)\overrightarrow{u}+(-1)\overrightarrow{v} \end{gathered}$$

Finding the coordinate vector of w→

Since we have

$$\begin{gathered} \overrightarrow{w}=-\overrightarrow{u}+-\overrightarrow{v} \\ \Rightarrow \overrightarrow{w}=\left(-1\right)\overrightarrow{u}+\left(-1\right)\overrightarrow{v} \\ \Rightarrow {\left[\overrightarrow{w}\right]}_{؏}=\left[\begin{array}{c}-1\\ -1\end{array}\right] \end{gathered}$$

Thus, the coordinate vector of$\overrightarrow{w}$ isrole="math" localid="1660379390895" ${\left[\overrightarrow{w}\right]}_{؏}=\left[\begin{array}{c}-1\\ -1\end{array}\right]\mathrm{Where}؏=\left\{\overrightarrow{u},\overrightarrow{v}\right\}$.

 Final Answer

If $؏=\left\{\overrightarrow{u},\overrightarrow{v}\right\}$be the basis and the vector $\overrightarrow{w}$is resultant of the two vector $\overrightarrow{u,}\overrightarrow{v}$then the coordinate vector of $\overrightarrow{w}$is given by-

${\left[\overrightarrow{w}\right]}_{؏}=\left[\begin{array}{c}-1\\ -1\end{array}\right]\mathrm{Where}؏=\left\{\overrightarrow{u},\overrightarrow{v}\right\}$