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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2:26 minutes

Problem 37

Textbook Question

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

Verified step by step guidance

Step 1: Understand the problem. We need to find the magnitude of the vector (w - u).

Step 2: Calculate the vector (w - u) by subtracting vector u from vector w.

Step 3: Substitute the given vectors: w = -i - 6j and u = 2i - 5j into the expression (w - u).

Step 4: Perform the subtraction: (-i - 6j) - (2i - 5j) to find the components of the resulting vector.

Step 5: Use the formula for the magnitude of a vector ||a|| = \sqrt{a_x^2 + a_y^2} to find ||w - u||.

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Key Concepts

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

Vector Subtraction

Vector subtraction involves finding the difference between two vectors by subtracting their corresponding components. For vectors u and w, this means subtracting the i and j components separately: (w_x - u_x)i + (w_y - u_y)j. This operation results in a new vector that represents the direction and magnitude from vector u to vector w.

Magnitude of a Vector

The magnitude of a vector, denoted as ||v||, is a measure of its length in space. For a vector expressed as v = ai + bj, the magnitude is calculated using the formula ||v|| = √(a² + b²). This concept is essential for understanding how far a vector extends from the origin to its endpoint in a Cartesian coordinate system.

Vector Notation and Components

Vectors are typically represented in component form as a combination of unit vectors, such as i and j in two dimensions. The components indicate the vector's influence in the horizontal (i) and vertical (j) directions. Understanding this notation is crucial for performing operations like addition, subtraction, and finding magnitudes, as it allows for clear manipulation of the vector's properties.