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 17c

Chapter 3, Problem 17c

Let E = 2i + 3j and F = 2i - 2j. Find the magnitude of -E - 2F.

Verified step by step guidance

Step 1: Write down the given vectors. E = 2i + 3j and F = 2i - 2j.

Step 2: Compute the expression -E. To negate a vector, multiply each component by -1. This gives -E = -2i - 3j.

Step 3: Compute the expression -2F. Multiply each component of F by -2. This gives -2F = -4i + 4j.

Step 4: Add the vectors -E and -2F. Combine the i-components and j-components separately: (-2i - 3j) + (-4i + 4j) = (-2 - 4)i + (-3 + 4)j = -6i + 1j.

Step 5: Find the magnitude of the resulting vector -6i + 1j. Use the formula for the magnitude of a vector: |V| = sqrt((x-component)^2 + (y-component)^2). Substitute the components: |V| = sqrt((-6)^2 + (1)^2).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

Vector Addition and Subtraction

Vector addition and subtraction involve combining vectors by adding or subtracting their corresponding components. For example, if vector E has components (2, 3) and vector F has components (2, -2), then -E would be (-2, -3) and -2F would be (-4, 4). The resultant vector is found by summing these component-wise.

Magnitude of a Vector

The magnitude of a vector is a measure of its length and is calculated using the formula √(x² + y²) for a 2D vector with components (x, y). This value represents the distance from the origin to the point defined by the vector in the Cartesian plane. For example, the magnitude of vector (a, b) is √(a² + b²).

Scalar Multiplication of Vectors

Scalar multiplication involves multiplying a vector by a scalar (a real number), which scales the vector's magnitude without changing its direction. For instance, multiplying vector F by -2 scales its components by -2, effectively reversing its direction and doubling its length. This operation is crucial for manipulating vectors in various physics problems.

Recommended video:

Guided course

04:06

Introduction to Vectors and Scalars

Related Practice

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.

views

Textbook Question

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

views

Textbook Question

Let A = 4i - 2j, B = -3i + 5j, and F = A - 4B. Draw a coordinate system and on it show vectors A, B, and F.

views

Textbook Question

Let E = 2i + 3j and F = 2i - 2j. Find the magnitude of E and F.

Textbook Question

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

Textbook Question

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

views