Vector 𝐀 has a magnitude of 12.3 units and points due west. Vector 𝐁 points due north. (a) What

step 1 Welcome back for this problem. We have two vectors. We'll close first vector vector A. This litt

Vector 𝐀 has a magnitude of 12.3 units and points due west....

Vector $\overrightarrow{\mathbf{A}}$ has a magnitude of 12.3 units and points due west. Vector $\overrightarrow{\mathbf{B}}$ points due north. (a) What is the magnitude of $\overrightarrow{\mathbf{B}}$ if $\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}$ has a magnitude of 15.0 units? (b) What is the direction of $\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}$ relative to due west? (c) What is the magnitude of $\overrightarrow{\mathbf{B}}$ if $\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}$ has a magnitude of 15.0 units? (d) What is the direction of $\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}$ relative to due west?

Key Concepts

Vector Magnitude

Vector magnitude represents the length or size of a vector and is calculated using the square root of the sum of the squares of its components. This concept is fundamental in translating physical dimensions to a scalar quantity that reflects the overall impact of the vector in space.

Vector Direction

Vector direction refers to the orientation of a vector relative to a reference axis, typically expressed in degrees or radians. Determining the direction is crucial for understanding how the vector's effect is distributed across different axes.

Vector Addition and Subtraction

Vector addition and subtraction are methods used to combine or differentiate effects represented by different vectors, typically by adding or subtracting their respective components. These operations enable the determination of a resultant vector when multiple forces or movements interact.

Component Decomposition

Component decomposition involves breaking a vector into its horizontal and vertical (or generally orthogonal) components. This process is essential for simplifying vector calculations, particularly when vectors are oriented in different directions.

Pythagorean Theorem in Vector Analysis

The Pythagorean theorem is used to calculate the magnitude of a resultant vector when its components are perpendicular to each other. By applying the theorem, one can determine the overall size of the vector from its individual components.

Trigonometric Functions in Vector Analysis

Trigonometric functions, such as tangent and arctangent, are employed to calculate the angle or direction of a resultant vector from its components. This mathematical tool allows the conversion between component form and angular direction in the analysis of vectors.

Transcript

00:01 Welcome back for this problem.

00:03 We have two vectors.

00:05 We'll close first vector vector a.

00:10 This little arrow above the a denotes that it's a vector.

00:15 It has both magnitude and a direction and vector b.

00:21 All right, and those are going to form a resultant vector of length 15 units.

00:32 We know that vector a is 12 .3 units.

00:39 All right, so there's our problem right there.

00:41 So we need to first find the length of or magnitude of vector b.

00:48 So using the pythagrin theorem, you can say that 15 squared is equal to b squared plus a squared, which is 12 .3 units.

00:59 You can subtract 12 .3 from both sides and then take the square root of both sides.

01:11 Square root, square root, and that will give us that b is equal to 8 .6 units.

01:22 And we need to also find the direction of the resultant vector.

01:29 So we're going to find this angle theta right here.

01:32 We can do that.

01:35 Theta is equal to the arc tangent of the side opposite of angle theta, which is side b, which we found earlier was 8 .6 units over 12 .3 units.

01:57 So theta is going to be equal to 34 .9 degrees.

02:05 All right, there's the first part of our, first excuse me, two parts of our problem...

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