Question

In Figurethree vectors are represented by arrows in the xyplane. Each square in the grid represents one meter. For each vector, write out the components of the vector, and calculate the magnitude of the vector.

Figure 1.56

Short answer

The components and magnitude of the vectors are

Step-by-step solution

Definition of magnitude of a vector

Magnitude of a vector $\stackrel{\rightharpoonup }{a}$having coordinates is given by $\left|\stackrel{\rightharpoonup }{a}\right|=\sqrt{x^2+y^2+z^2}$.

The magnitude of a vector gives the length of the vector so, basically magnitude is used to find the length of a vector between the initial and the final point of the vector.

Vector components of given vectors

Any vector directed in two dimensions can be thought of as having an influence in two different directions. That is, it can be thought of as having two parts. Each part of a two-dimensional vector is known as a component.

The components of a vector depict the influence of that vector in a given direction.

Write the vector $\stackrel{\rightharpoonup }{a}$, $\stackrel{\rightharpoonup }{c}$, and $\stackrel{\rightharpoonup }{g}$in the component form from the given figure.

Calculation of magnitude

Substitute x=5, y=3 and z=0 into the formula of magnitude to get the magnitude of the vector $\stackrel{\rightharpoonup }{a}$.

$$\begin{gathered} \left|\stackrel{\rightharpoonup }{a}\right|=\sqrt{{\left(5\right)}^2+{\left(3\right)}^2+{\left(0\right)}^2} \\ =5.83m \end{gathered}$$

Substitute x=6, y=-9 and z=0 into the formula of magnitude to get the magnitude of the vector $\stackrel{\rightharpoonup }{c}$.

$$\begin{gathered} \left|\stackrel{\rightharpoonup }{c}\right|=\sqrt{{\left(6\right)}^2+{\left(-9\right)}^2+{\left(0\right)}^2} \\ =10.82m \end{gathered}$$

Substitute x=-10, y=3 and z=0 and into the formula of magnitude to get the magnitude of the vector $\stackrel{\rightharpoonup }{g}$.

$$\begin{gathered} \left|\stackrel{\rightharpoonup }{g}\right|=\sqrt{{\left(-10\right)}^2+{\left(3\right)}^2+{\left(0\right)}^2} \\ =10.44m \end{gathered}$$

Thus, the components and magnitude of the vectors are

and$\left|\stackrel{\rightharpoonup }{c}\right|=10.44m.$