Question

The displacement vector from your house to the library is \(760 \mathrm{~m}\) long, pointing \(35^{\circ}\) north of east. What are the components of this displacement vector?

Short answer

The displacement components are approximately 622.6 m east and 435.9 m north.

Step-by-step solution

Understanding Vector Components

A vector can be broken down into horizontal (east-west) and vertical (north-south) components based on its magnitude and direction. In this case, the vector has a magnitude of 760 m and forms a 35° angle north of east.

Finding the Horizontal Component

The horizontal (east) component of the vector can be calculated using the cosine function: \[ \text{Horizontal component} = 760 \times \cos(35^{\circ}) \] Calculate this value using a calculator that is in degree mode.

Calculating the Horizontal Component

Using a calculator, compute the horizontal component: \[ \text{Horizontal component} = 760 \times \cos(35^{\circ}) \approx 760 \times 0.8192 \approx 622.6 \text{ m} \]. Thus, the horizontal component is approximately 622.6 meters east.

Finding the Vertical Component

The vertical (north) component of the vector can be calculated using the sine function: \[ \text{Vertical component} = 760 \times \sin(35^{\circ}) \] Calculate this value using a calculator that is in degree mode.

Calculating the Vertical Component

Using a calculator, compute the vertical component: \[ \text{Vertical component} = 760 \times \sin(35^{\circ}) \approx 760 \times 0.5736 \approx 435.9 \text{ m} \]. Thus, the vertical component is approximately 435.9 meters north.

Summarize the Components

The displacement vector from the house to the library has components approximately 622.6 meters east and 435.9 meters north.

Key concepts

Displacement

Displacement refers to the change in position of an object. It's not just about the distance covered but also the direction of movement. Unlike distance, which is a scalar quantity and has only magnitude, displacement is a vector quantity and has both magnitude and direction.

For example, if you move from your house to the library, which is 760 meters away and 35 degrees north of east, the direct path you take between these two locations is your displacement vector. Here, the 760 meters denotes how far you moved, and the angle of 35 degrees indicates the direction of the motion.

Always remember:

• Displacement is straight-line distance between start and end points, considering direction.
• Magnitude is the length of the vector, measured in meters in our example.
• Direction is specified by the angle from a reference direction, like north of east in this case.

Components of a Vector

Understanding the components of a vector is crucial in physics and mathematics. A vector can be decomposed into two perpendicular components, usually base on a coordinate system, such as the horizontal (east-west) and vertical (north-south) components. This helps simplify the problem and makes it easier to analyze vector quantities.

When you decompose a displacement vector, as seen in the exercise example, you use trigonometric functions:

• Use the cosine function to find the horizontal component: \[ \text{Horizontal component} = \text{Magnitude} \times \cos(\text{angle}) \]. In our case, it is calculated as \( 760 \times \cos(35^{\circ}) \approx 622.6 \) meters east.
• Use the sine function to find the vertical component: \[ \text{Vertical component} = \text{Magnitude} \times \sin(\text{angle}) \]. For our exercise, it results in \( 760 \times \sin(35^{\circ}) \approx 435.9 \) meters north.

By breaking the vector into components, we can better understand its effect in two perpendicular directions, making it simpler to work with.

Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles. It's particularly useful in breaking down vectors in physics. When you have a vector, like a displacement vector from a point in a certain direction, trigonometry allows you to find its components along different axes.

Key trigonometric functions used in vector decomposition include:

• Cosine (cos) relates the adjacent side (horizontal component) to the hypotenuse (magnitude of vector). For an angle \( \theta \), \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \).
• Sine (sin) relates the opposite side (vertical component) to the hypotenuse (magnitude of vector). For an angle \( \theta \), \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \).

In our example, with an angle of 35 degrees north of east, we use:

• The cosine function to find the horizontal (east) component: \( 760 \times \cos(35^{\circ}) \).
• The sine function to find the vertical (north) component: \( 760 \times \sin(35^{\circ}) \).

By understanding these relationships, you can easily apply trigonometry to solve problems involving vectors.