Table of contents

0. Review of College Algebra4h 43m
1. Measuring Angles40m
2. Trigonometric Functions on Right Triangles2h 5m
3. Unit Circle1h 19m
4. Graphing Trigonometric Functions1h 19m
5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
6. Trigonometric Identities and More Equations2h 34m
7. Non-Right Triangles1h 38m
8. Vectors2h 25m
9. Polar Equations2h 5m
10. Parametric Equations1h 6m
11. Graphing Complex Numbers1h 7m

8. Vectors

Geometric Vectors

Trigonometry

8. Vectors

Geometric Vectors

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1:40 minutes

Problem 29

Textbook Question

In Exercises 21–38, letu = 2i - 5j, v = -3i + 7j, and w = -i - 6j.Find each specified vector or scalar.-4w

Verified step by step guidance

Identify the vector w, which is given as \(-\mathbf{i} - 6\mathbf{j}\).

To find \(-4\mathbf{w}\), multiply each component of the vector \(\mathbf{w}\) by \(-4\).

Multiply the \(\mathbf{i}\) component: \(-4 \times (-1)\).

Multiply the \(\mathbf{j}\) component: \(-4 \times (-6)\).

Combine the results to express the vector \(-4\mathbf{w}\) in terms of \(\mathbf{i}\) and \(\mathbf{j}\).

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Key Concepts

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

Vector Operations

Vector operations include addition, subtraction, and scalar multiplication. In this context, scalar multiplication involves multiplying a vector by a scalar (a real number), which scales the vector's magnitude while maintaining its direction. For example, multiplying a vector by -4 will reverse its direction and increase its length by a factor of 4.

Vector Representation

Vectors in two-dimensional space can be represented in the form of 'ai + bj', where 'a' and 'b' are the components along the x-axis and y-axis, respectively. In this question, the vectors u, v, and w are expressed in this format, allowing for straightforward manipulation and calculation of their properties.

Scalar Multiplication of Vectors

Scalar multiplication of a vector involves multiplying each component of the vector by the scalar. For instance, if we have a vector w = -i - 6j and we multiply it by -4, we calculate -4 * (-1) for the i component and -4 * (-6) for the j component, resulting in a new vector that reflects the scaling effect of the scalar.