Question

In Exercises 75 and \(76,\) sketch the vector \(v\) and write its component form. \(\mathbf{v}\) lies in the \(y z\) -plane, has magnitude 2 , and makes an angle of \(30^{\circ}\) with the positive \(y\) -axis.

Short answer

The vector \(\mathbf{v}\) in component form is \(\mathbf{v} = (0, \sqrt{3}, 1)\).

Step-by-step solution

Drawing the Vector

Start by drawing a coordinate system for the \(y z\)-plane. Then, sketch a vector \(\mathbf{v}\) that makes an angle of \(30^{\circ}\) with the positive \(y\)-axis. The vector will point up and to the right. Remember, the magnitude of vector \(\mathbf{v}\) is 2, but for now, we just want to get a sense of direction.

Calculating the Components

To find the components of the vector, we rely on its magnitude and the angle it makes with the positive \(y\)-axis. Considering \(\mathbf{v}\) as the hypotenuse of a right triangle, the \(y\)-component (adjacent to the angle) will be \(2 \times \cos(30^{\circ})\) and the \(z\)-component (opposite to the angle) will be \(2 \times \sin(30^{\circ})\). Converting the degrees to radians, \(30^{\circ}\) equates to \(30 \times (\pi / 180) = \pi / 6\). Then, the \(y\)-component equals \(2 \times \cos(\pi / 6)\) = \(2 \sqrt{3} / 2 = \sqrt{3}\). The \(z\)-component equals \(2 \times \sin(\pi / 6)\) = \(2 / 2 = 1\). Hence, the vector in its component form will be \(\mathbf{v} = (0, \sqrt{3}, 1)\). Note that the \(x\) -component is 0 because the vector \(\mathbf{v}\) lies in the \(y z\) -plane and is perpendicular to the \(x\)-axis.

Final Answer

Combine the calculated components to form vector \(\mathbf{v}\). Therefore, in component form, the vector \(\mathbf{v}\) is \(\mathbf{v} = (0, \sqrt{3}, 1)\).

Key concepts

Magnitude of a Vector

Understanding the magnitude of a vector is crucial in physics and mathematics as it represents the vector's length or size. In simple terms, the magnitude is the distance from the vector's initial point to its terminal point, regardless of its direction. To calculate the magnitude of a vector, we use the Pythagorean theorem for a vector with components in the two-dimensional plane or its three-dimensional extension for vectors in space.
For example, for a vector \( \mathbf{v} \) with components \( (x, y, z) \), the magnitude \( |\mathbf{v}| \) is found using the formula \( |\mathbf{v}| = \sqrt{x^2 + y^2 + z^2} \). In the case of our exercise, the vector \( \mathbf{v} \) has a given magnitude of 2. Understanding how the magnitude relates to the vector's components helps to derive these components based on the magnitude and the angles the vector makes with the coordinate axes.

Trigonometry in Vectors

The use of trigonometry in vectors allows us to break down a vector into its constituent components along the axes of a coordinate system. This is particularly useful when dealing with vectors at an angle to the standard coordinate axes. Trigonometric functions such as sine and cosine are essential in relating the angle of the vector with its components.

Breaking Down a Vector Using Sine and Cosine

If a vector forms an angle with one of the coordinate axes, we can use the sine and cosine of that angle to find the vector's components. For a vector in a plane, the component along the axis with which it forms an angle is found by multiplying the vector's magnitude by the cosine of the angle, while the component along the perpendicular axis is found by multiplying by the sine of the angle.
In the provided step-by-step solution, the \(y\)-component of vector \(\mathbf{v}\) is obtained using \(2 \times \cos(30^{\bf\text{\textdegree}})\), and the \(z\)-component by \(2 \times \sin(30^{\bf\text{\textdegree}})\). This process enables us to calculate the exact components of the vector when it is not aligned with the primary axes.

Coordinate System

A coordinate system is a framework that allows us to locate points in space using a set of numbers known as coordinates. It consists of fixed reference lines or axes and a set of rules that uses these axes to define the position of any point or geometric element.

Understanding the Cartesian Coordinate System

The most commonly used system in mathematics is the Cartesian coordinate system, which is defined by perpendicular axes—commonly labeled as the x, y, and z axes in three-dimensional space. In our exercise, the vector lies in the \(yz\)-plane, indicating that it does not have any component in the direction of the \(x\)-axis, hence the \(x\)-component is zero.
Vectors can be represented in this coordinate system by their components along the respective axes. When working on problems like the exercise presented, it's essential to visualize the vector's direction concerning these axes to correctly calculate its components. By combining knowledge of trigonometry with the coordinate system, we can describe any vector's position and direction.