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 10c

Chapter 3, Problem 10c

Draw each of the following vectors, label an angle that specifies the vector's direction, and then find the vector's magnitude and direction. v = (14i - 11j) m/s

Verified step by step guidance

Step 1: Understand the vector components. The vector v is given as v = (14i - 11j) m/s, where 14i represents the x-component (horizontal direction) and -11j represents the y-component (vertical direction).

Step 2: Draw the vector on a coordinate system. Start at the origin (0, 0). Move 14 units to the right along the x-axis (positive direction) and then move 11 units downward along the y-axis (negative direction). Label the vector v and the angle θ it makes with the positive x-axis.

Step 3: Calculate the magnitude of the vector using the Pythagorean theorem. The magnitude |v| is given by the formula:

\sqrt{14^{2} + 11^{2}}

. Substitute the values and simplify.

Step 4: Determine the direction of the vector. The angle θ can be calculated using the formula:

θ = \tan - 1^{-1} (\frac{- 11}{14})

. Note that the negative sign in the y-component indicates the vector points downward.

Step 5: Interpret the angle θ. Since the vector lies in the fourth quadrant (positive x-component and negative y-component), adjust the angle if necessary to ensure it is measured counterclockwise from the positive x-axis. Express the direction in degrees or radians as appropriate.

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 Representation

Vectors are quantities that have both magnitude and direction, represented in a coordinate system. In this case, the vector v = (14i - 11j) m/s can be visualized in a two-dimensional plane, where 'i' represents the x-component and 'j' represents the y-component. Understanding how to graphically represent vectors is essential for visualizing their direction and magnitude.

Magnitude of a Vector

The magnitude of a vector is a measure of its length, calculated using the Pythagorean theorem. For the vector v = (14i - 11j) m/s, the magnitude is found by taking the square root of the sum of the squares of its components: |v| = √(14² + (-11)²). This value provides a scalar quantity that represents the strength of the vector.

Direction of a Vector

The direction of a vector is often specified by the angle it makes with a reference axis, typically the positive x-axis. This angle can be calculated using the arctangent function: θ = arctan(y/x), where y and x are the vector's components. For the vector v = (14i - 11j) m/s, determining the angle helps in understanding how the vector is oriented in the coordinate system.

Recommended video:

Guided course

06:44

Adding 3 Vectors in Unit Vector Notation

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. B = -4.0i + 4.0j

views

Textbook Question

A runner is training for an upcoming marathon by running around a 100-m-diameter circular track at constant speed. Let a coordinate system have its origin at the center of the circle with the x-axis pointing east and the y-axis north. The runner starts at (x,y) = (50m, 0m) and runs 2.5 times around the track in aclockwise direction. What is his displacement vector? Give your answer as a magnitude and direction.

views

Textbook Question

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

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

views

Textbook Question

Draw each of the following vectors, label an angle that specifies the vector's direction, and then find the vector's magnitude and direction. A = 3.0i + 7.0j

views

Textbook Question

Let C = (3.15 m, 15 degrees above the neagtive x-axis) and D = (25.6, 30 degrees to the right of the negative y-axis). Find the x and y components of each vector.

views

Textbook Question

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

views