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Chapter 6: Problem 101

Explain how to write a vector in terms of its magnitude and direction.

Short Answer

Expert verified
A vector can be written in terms of its magnitude and direction as follows: magnitude is calculated as the square root of the sum of the squares of the vector's components (\( \sqrt{x^2 + y^2} \)), and the direction is the arctangent of the ratio of the y and x components (\( tan^{-1}\frac{y}{x} \)).

Step by step solution

01

Identify the vector

You must first determine the vector you are working with. A vector is typically represented as \( \vec{V} = x\vec{i} + y\vec{j} \) in a two-dimensional space, where \(\vec{i}\) and \(\vec{j}\) are the unit vectors along the x and y axes, respectively, and 'x' and 'y' are the vector's components.
02

Calculate the magnitude

Next, the magnitude of the vector is calculated. This can be done using the Pythagorean theorem: \( |\vec{V}| = \sqrt{x^2 + y^2} \). This represents the length of the vector.
03

Determine the direction

Finally, to find the direction (胃) of the vector, you use the tangent function: \( tan(\theta) = \frac{y}{x} \). Here, 'y' is the vertical component of the vector and 'x' is the horizontal component. The resulting angle is typically measured in degrees or radians and represents the direction of the vector from the positive x-axis.

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