Ch 03: Vectors and Coordinate Systems

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 03: Vectors and Coordinate Systems

Problem 12c

Chapter 3, Problem 12c

Let A = 4i - 2j, B = -3i + 5j, and C = A + B. What are the magnitude and direction of vector C?

Verified step by step guidance

Step 1: Add the components of vectors A and B to find vector C. The x-component of C is the sum of the x-components of A and B: $ C_x = A_x + B_x $. Similarly, the y-component of C is the sum of the y-components of A and B: $ C_y = A_y + B_y $.

Step 2: Substitute the given components of A and B into the equations. For A = 4i - 2j, $ A_x = 4 $ and $ A_y = -2 $. For B = -3i + 5j, $ B_x = -3 $ and $ B_y = 5 $. Calculate $ C_x $ and $ C_y $.

Step 3: Use the Pythagorean theorem to calculate the magnitude of vector C. The formula is $ |C| = \sqrt{C_x^2 + C_y^2} $. Substitute the values of $ C_x $ and $ C_y $ into this formula.

Step 4: Determine the direction (angle) of vector C using the formula $ \theta = \arctan\left(\frac{C_y}{C_x}\right) $. Ensure you consider the signs of $ C_x $ and $ C_y $ to determine the correct quadrant for the angle.

Step 5: Express the final answers for the magnitude and direction of vector C. The magnitude is a scalar value, and the direction is typically expressed in degrees or radians, measured counterclockwise from the positive x-axis.

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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 two or more vectors to form a resultant vector. This is done by adding their corresponding components. For example, if vector A has components (4, -2) and vector B has components (-3, 5), the resultant vector C is obtained by adding these components: C = (4 + (-3), -2 + 5) = (1, 3).

Magnitude of a Vector

The magnitude of a vector is a measure of its length and is calculated using the Pythagorean theorem. For a vector C with components (x, y), the magnitude is given by |C| = √(x² + y²). This value represents how far the vector extends in space, regardless of its direction.

Direction of a Vector

The direction of a vector indicates the angle it makes with a reference axis, typically the positive x-axis. It can be expressed in degrees or radians and is often calculated using the tangent function: θ = arctan(y/x), where (x, y) are the components of the vector. This information is crucial for understanding how the vector is oriented in space.

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Related Practice

Textbook Question

Draw each of the following vectors, label an angle that specifies the vector's direction, then find its magnitude and direction.

$\vec{a} = (20 i + 10 j) {m/s}^{2}$

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Textbook Question

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Draw each of the following vectors, label an angle that specifies the vector's direction, then find its magnitude and direction. r = (-2.0i - 1.0j) cm

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Let A = 2i + 3j, B = 2i - 4j, and C = A + B. Draw a coordinate system and on it show vectors A, B, and C.

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Textbook Question

Let A = 4i - 2j, B = -3i + 5j, and C = A + B. Write vector C in component form.

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