Problem 14 - College Trigonometry

Question

Find the component form of the vector \(\vec{v}\) using the information given about its magnitude and direction. Give exact values. \(\|\vec{v}\|=12\); when drawn in standard position \(\vec{v}\) lies along the positive \(y\) -axis

Step-by-Step Solution

Answer

The component form of \( \vec{v} \) is \( \langle 0, 12 \rangle \).

Step 1: Understand the Problem

We are asked to find the component form of a vector \( \vec{v} \) given its magnitude is 12, and it lies along the positive \( y \)-axis when drawn in standard position.

Step 2: Recognize Vector Direction

Since the vector is along the positive \( y \)-axis, its direction makes a 90-degree angle with the positive \( x \)-axis. In component form, this means the \( x \)-component is 0, and there is only a \( y \)-component.

Step 3: Use Magnitude to Find Components

Given the vector is entirely along the positive \( y \)-axis, the \( y \)-component must equal the magnitude of the vector, which is given as 12.

Step 4: Write the Component Form of the Vector

The component form of \( \vec{v} \) with \( x \)-component 0 and \( y \)-component 12 is \( \vec{v} = \langle 0, 12 \rangle \).

Key Concepts

Magnitude of a Vector

The magnitude of a vector represents the length or size of the vector. To find it, you use the formula: \[\| \vec{v} \| = \sqrt{x^2 + y^2} \]This works for any vector expressed in component form \(\langle x, y \rangle\).It's like using the Pythagorean theorem in geometry.However,if a vector is along a specific axis, the magnitude is simply equal to the non-zero component.For instance, a vector \(\vec{v}\) with component form \(\langle 0, 12 \rangle\) has a magnitude of 12 because all its length lies along the y-axis.

Direction of a Vector

A vector's direction is often thought of as the angle it makes with the positive x-axis. For vectors in a plane,you can use trigonometry to find this angle.If \(\theta\) represents the angle, you calculate it using:\[ \theta = \tan^{-1} \left( \frac{y}{x} \right) \]However,when a vector lies along an axis, determining the direction becomes straightforward.If a vector \(\vec{v}\) lies along the positive y-axis, its direction is a 90-degree angle.Similarly,if it lies along the x-axis,it points either at 0 degrees (positive direction)or 180 degrees (negative direction).

Standard Position

Vectors are said to be in standard position when their initial point is at the origin,\((0,0)\), on a coordinate plane.This simplifies analyzing and comparing vectors. In the standard position,one describes the vector solely by its terminal point,which is where the vector's point ends when it starts from the origin. For example,a vector with terminal point at \((0,12)\) has a clear path along the positive y-axis. This is helpful when solving problems as it aligns the component form with the physical direction of the vector.