Problem 37 - Precalculus Mathematics for Calculus

Question

Find the work done by the force \(\mathbf{F}\) in moving an object from \(P\) to \(Q\). $$\mathbf{F}=10 \mathbf{i}+3 \mathbf{j} ; \quad P(2,3), Q(6,-2)$$

Step-by-Step Solution

Answer

The work done by the force is 25 units.

Step 1: Determine the Displacement Vector

To find the displacement vector \( \mathbf{d} \) from point \( P(2,3) \) to \( Q(6,-2) \), we calculate the difference between the coordinates of \( Q \) and \( P \). The displacement vector \( \mathbf{d} \) is given by: \[ \mathbf{d} = (6 - 2)\mathbf{i} + (-2 - 3)\mathbf{j} = 4\mathbf{i} - 5\mathbf{j} \]

Step 2: Understand the Dot Product

The work done by a force is calculated using the dot product of the force vector \( \mathbf{F} \) and the displacement vector \( \mathbf{d} \). The dot product formula is: \[ \mathbf{F} \cdot \mathbf{d} = F_x d_x + F_y d_y \] where \( F_x \) and \( F_y \) are components of \( \mathbf{F} \), and \( d_x \) and \( d_y \) are components of \( \mathbf{d} \).

Step 3: Calculate the Dot Product

Calculate the dot product using the force vector \( \mathbf{F} = 10 \mathbf{i} + 3 \mathbf{j} \) and the displacement vector \( \mathbf{d} = 4 \mathbf{i} - 5 \mathbf{j} \):\[ \mathbf{F} \cdot \mathbf{d} = (10)(4) + (3)(-5) = 40 - 15 = 25 \]

Step 4: Interpret the Result

The result of the dot product tells us the work done by the force \( \mathbf{F} \). In this case, the work done is 25 units.

Key Concepts

Displacement Vector

A displacement vector is a crucial concept in physics and vector mathematics. It is a vector that represents the change in position of an object from one point to another. Think of it as a directional arrow pointing from an initial position to a final position.
The displacement vector is calculated by subtracting the initial position coordinates from the final position coordinates. In simple terms, if you imagine you are walking from one point to another, the displacement vector shows you the straight path you have traveled, regardless of the path you actually took.

• The formula for finding the displacement vector \( \mathbf{d} \) when moving from point \( P(x_1, y_1) \) to point \( Q(x_2, y_2) \) is: \[ \mathbf{d} = (x_2 - x_1)\mathbf{i} + (y_2 - y_1)\mathbf{j} \]
• In our example, from \( P(2,3) \) to \( Q(6,-2) \), we calculated: \( \mathbf{d} = 4\mathbf{i} - 5\mathbf{j} \)

Dot Product

The dot product, also known as the scalar product, is a way to multiply two vectors, resulting in a single number or scalar. This scalar is particularly useful in determining things like work done in physics because it considers both the magnitude of the vectors and the angle between them.

• The dot product of two vectors \( \mathbf{A} = a_x\mathbf{i} + a_y\mathbf{j} \) and \( \mathbf{B} = b_x\mathbf{i} + b_y\mathbf{j} \) is calculated as: \[ \mathbf{A} \cdot \mathbf{B} = a_x b_x + a_y b_y \]
• In the context of work done, the dot product tells us how much of the force is applied in the direction of the displacement.

In our example, with the force vector \( \mathbf{F} = 10\mathbf{i} + 3\mathbf{j} \) and the displacement vector \( \mathbf{d} = 4\mathbf{i} - 5\mathbf{j} \), the work done \( \mathbf{F} \cdot \mathbf{d} \) is computed as 25 units.

Vector Components

Understanding vector components is essential to solving problems involving forces and motion. Any vector in a plane can be broken down into its components along the common x and y axes. This makes it easier to calculate properties like displacement or force in each direction.
By using coordinate axes, you can represent complex movements or forces in simpler, linear forms.

• A vector \( \mathbf{V} \) with an \( x \)-component \( V_x \) and a \( y \)-component \( V_y \) can be expressed as: \[ \mathbf{V} = V_x\mathbf{i} + V_y\mathbf{j} \]
• In our scenario, the force vector \( \mathbf{F} = 10\mathbf{i} + 3\mathbf{j} \) has components 10 in the \( \mathbf{i} \)-direction (x-axis) and 3 in the \( \mathbf{j} \)-direction (y-axis).

These components are used in calculations to ensure alignment with the relevant axes, making complex calculations manageable.