Question

Sketch the vector \(v\) and write its component form. \(\mathbf{v}\) lies in the \(x z\) -plane, has magnitude \(5,\) and makes an angle of \(45^{\circ}\) with the positive \(z\) -axis.

Short answer

The component form of vector \(v\) is \(\left(\frac{5\sqrt{2}}{2}, 0, \frac{5\sqrt{2}}{2}\right)\).

Step-by-step solution

Visualizing the Vector

First, visualize the vector \(v\). We know it lies in the \(x z\) -plane. As the angle description is given relative to the \(z\)-axis, we will treat the \(x\)-axis as horizontal and the \(z\)-axis as vertical. The magnitude (length) of \(v\) is \(5\). This vector makes a \(45^{\circ}\) angle with the positive \(z\)-axis.

Direction of Vector

Next, determine the direction of the vector. Based on the information, we only know that it makes a \(45^{\circ}\) angle with the \(z\)-axis, but whether it makes an angle with the \(x\)-axis is not mentioned. Because it's on the \(xz\)-plane, the vector could be projected onto the \(x\)-axis or the \(z\)-axis. By convention, we can assume that the vector can be projected onto the \(x\)-axis. Therefore, the vector also makes a \(45^{\circ}\) angle with the \(x\)-axis.

Finding the Components

Now, find the components of the vector. We can use the angle it makes with the \(x\)-axis (\(45^{\circ}\)) and its magnitude (5) to determine its components. The \(x\)-component of the vector is therefore \(5 \cos(45^{\circ})\) and the \(z\)-component is \(5 \sin(45^{\circ})\). As the vector lies in the \(xz\)-plane, the \(y\)-component is \(0\).

Calculating the Exact Values

Calculate the exact values. Given that \(\cos(45^{\circ}) = \sin(45^{\circ}) = \frac{\sqrt{2}}{2}\), the \(x\)- and \(z\)-components of the vector become \(5 \frac{\sqrt{2}}{2} = \frac{5\sqrt{2}}{2}\) and its \(y\)-component remains \(0\).