Question

a. How does the result of Exercise 52 change if the force due to the wind is \(\mathbf{F}=\langle 141,50\rangle\) (approximately the same magnitude, but different direction)? b. How does the result of Exercise 52 change if the force due to the wind is \(\mathbf{F}=\langle 141,-50\rangle\) (approximately the same magnitude, but different direction)?

Short answer

Answer: The total force on the sailboat is \(\mathbf{T}=\langle 189,50 \rangle\) in the first case and \(\mathbf{T}=\langle 189,-50 \rangle\) in the second case.

Step-by-step solution

Case 1 - Force due to the wind: \(\mathbf{F}=\langle 141,50\rangle\)

The force due to the wind has been changed in this case to \(\mathbf{F}=\langle 141,50\rangle\). According to Exercise 52, the given force due to water current is \(\mathbf{W}=\langle 48,0 \rangle\). Now, in order to get the total force on the sailboat, we need to add both the wind force and water force vectors. So, we will carry out the vector addition.

Vector Addition

Adding the wind force vector \(\mathbf{F}\) and the water force vector \(\mathbf{W}\), we get: \(\mathbf{T} = \mathbf{F} + \mathbf{W} = \langle 141,50 \rangle + \langle 48,0 \rangle = \langle 141+48,50+0 \rangle = \langle 189,50 \rangle\) The total force on the sailboat in this case is \(\mathbf{T}=\langle 189,50 \rangle\).

Case 2 - Force due to the wind: \(\mathbf{F}=\langle 141,-50\rangle\)

The force due to the wind has been changed in this case to \(\mathbf{F}=\langle 141,-50\rangle\). According to Exercise 52, the given force due to water current is \(\mathbf{W}=\langle 48,0 \rangle\). Now, in order to get the total force on the sailboat, we need to add both the wind force and water force vectors. So, we will carry out the vector addition.

Vector Addition

Adding the wind force vector \(\mathbf{F}\) and the water force vector \(\mathbf{W}\), we get: \(\mathbf{T} = \mathbf{F} + \mathbf{W} = \langle 141,-50 \rangle + \langle 48,0 \rangle = \langle 141+48,-50+0 \rangle = \langle 189,-50 \rangle\) The total force on the sailboat in this case is \(\mathbf{T}=\langle 189,-50 \rangle\).

Key concepts

Vector Magnitude

Let's begin by understanding what vector magnitude means. In simple terms, the magnitude of a vector is a measure of its length or size. It is computed using the components of the vector itself. If a vector \( \mathbf{V} = \langle x,y \rangle \), its magnitude \( \|\mathbf{V}\| \) is calculated as:\[\|\mathbf{V}\| = \sqrt{x^2 + y^2}\]This formula is derived from the Pythagorean theorem, reflecting the fact that magnitude represents the hypotenuse of a right triangle formed by the vector components.

• For instance, if you have a vector \( \mathbf{F} = \langle 141, 50 \rangle \), you calculate its magnitude by plugging the values into the formula.
• This gives you: \( \|\mathbf{F}\| = \sqrt{141^2 + 50^2} \).
• Then solve this to find that \( \|\mathbf{F}\| \) is approximately 150.7.

By knowing the magnitude, you understand how strong or intense a vector quantity is, whether it’s about how windy it is or how strong a current flows in water.

Vector Components

Vectors are often easier to work with when broken down into their basic parts or components. These components reveal the influence of the vector in different directions. When dealing with a vector \( \mathbf{F} = \langle a, b \rangle \), \(a\) and \(b\) are its x-component and y-component respectively.

• The x-component represents the horizontal influence, telling us how much the vector affects the x-axis.
• The y-component shows the vertical influence, indicating the effect on the y-axis.

In the exercise given, changing the direction changes these components from \( \mathbf{F} = \langle 141, 50 \rangle \) to \( \mathbf{F} = \langle 141, -50 \rangle \). This switch means the force now pushes differently in the vertical direction, impacting results.

• Understanding components help visualize how forces interact in different scenarios. They are like the building blocks of vectors.
• They break down complex interactions into understandable parts.

Force Vectors

Force vectors are specifically used in physics to represent forces acting on an object. These vectors have both a magnitude, representing how strong the force is, and a direction, showing where the force is applied.

• In the exercise, forces such as wind and water current are modeled as vectors \( \mathbf{F} \) and \( \mathbf{W} \).
• Force vectors can be added together to find a resultant or total force vector. This process is known as vector addition.

To understand this better, imagine trying to sail a boat. If the wind's force is \( \mathbf{F} = \langle 141, 50 \rangle \) and the water exerts \( \mathbf{W} = \langle 48, 0 \rangle \), combining these forces shows the total effect on the boat—more wind means different sailing directions!

• Vector addition is performed by adding corresponding components of each vector: \(\mathbf{T} = \mathbf{F} + \mathbf{W} = \langle 141 + 48, 50 + 0 \rangle \).
• This gives a total force \( \mathbf{T} = \langle 189, 50 \rangle \) in one scenario, and \( \langle 189, -50 \rangle \) in another, signifying a push or pull in varying directions.