Let A=(3.0 m,20∘ south of east),B=(2.0 m,north)\mathbf{A} = (3.0 \, \text{m}, 20^\circ \, \text{south of east}), \quad \mathbf{B} = (2.0 \, \text{m}, \text{north}), and C=(5.0 m,70∘ south of west)\mathbf{C} = (5.0 \, \text{m}, 70^\circ \, \text{south of west}). Find the magnitude and the direction of D=A+B+C\mathbf{D} = \mathbf{A} + \mathbf{B} + \mathbf{C}.

Ch 03: Vectors and Coordinate Systems

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 03: Vectors and Coordinate Systems

Problem 22c

Chapter 3, Problem 22c

Let $A = (3.0 m, 20^{\circ} south of east),$ , and $C = (5.0 m, 70^{\circ} south of west)$ . Find the magnitude and the direction of $D = A + B + C$ .

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Break each vector into its components. For vector A, calculate the x-component as A_x = A * cos(20°) and the y-component as A_y = -A * sin(20°) (negative because it is south of east). For vector B, since it points directly north, its x-component is B_x = 0 and its y-component is B_y = B. For vector C, calculate the x-component as C_x = -C * cos(70°) (negative because it is west) and the y-component as C_y = -C * sin(70°) (negative because it is south).

Add the x-components of all three vectors to find the x-component of D: D_x = A_x + B_x + C_x.

Add the y-components of all three vectors to find the y-component of D: D_y = A_y + B_y + C_y.

Calculate the magnitude of vector D using the Pythagorean theorem: |D| = sqrt(D_x^2 + D_y^2).

Determine the direction of vector D by calculating the angle θ using the formula θ = arctan(D_y / D_x). Adjust the angle based on the signs of D_x and D_y to ensure it is in the correct quadrant.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Vector Addition

Vector addition involves combining multiple vectors to find a resultant vector. This process requires breaking each vector into its components, typically along the x (horizontal) and y (vertical) axes. The components are then summed separately to obtain the total x and y components of the resultant vector.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are essential for resolving vectors into their components based on their angles. For a vector at an angle θ, the x-component can be found using the cosine function (A_x = A * cos(θ)), and the y-component using the sine function (A_y = A * sin(θ)). This allows for accurate calculations of vector magnitudes and directions.

Magnitude and Direction of a Vector

The magnitude of a vector represents its length or size, while the direction indicates the angle at which it acts. After calculating the resultant vector's components, the magnitude can be found using the Pythagorean theorem (|D| = √(D_x² + D_y²)), and the direction can be determined using the arctangent function (θ = arctan(D_y / D_x)), which gives the angle relative to a reference axis.

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Textbook Question

Let $\vec{B}$ = (5.0 m, 30 degrees counterclockwise from vertically up). Find the x- and y-components of $\vec{B}$ in each of the two coordinate systems shown in FIGURE EX3.21.

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Textbook Question

The position of a particle as a function of time is given by $\vec{r}$ = ( 5.0î ＋4.0ĵ )t² m where t is in seconds. Find an expression for the particle's velocity $\vec{v}$ as a function of time.

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The position of a particle as a function of time is given by $\vec{r}$ = ( 5.0î ＋4.0ĵ )t² m where t is in seconds. What is the particle's distance from the origin at t = 0, 2, and 5 s?

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FIGURE EX3.19 shows vectors A and B. What is C = A + B? Write your answer in component form using unit vectors.

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What are the x- and y-components of the velocity vector shown in FIGURE EX3.20?

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