Problem 41 - College Trigonometry

Question

For the given vector \(\vec{v}\), find the magnitude \(\|\vec{v}\|\) and an angle \(\theta\) with \(0 \leq \theta<360^{\circ}\) so that \(\vec{v}=\|\vec{v}\|\langle\cos (\theta), \sin (\theta)\rangle\) (See Definition 11.8.) Round approximations to two decimal places. $$ \vec{v}=-10 \hat{\jmath} $$

Step-by-Step Solution

Answer

The magnitude is 10 and the angle \(\theta\) is \(270^{\circ}\).

Step 1: Identify the Vector Components

The vector \(\vec{v}\) is given as \(-10 \hat{\jmath}\). This represents a vector with no horizontal component (i.e., \(x = 0\)) and a vertical component of \(y = -10\).

Step 2: Calculate the Magnitude of the Vector

The magnitude of a vector \((x, y)\) is calculated using the formula \(\|\vec{v}\| = \sqrt{x^2 + y^2}\). Here, \(x = 0\) and \(y = -10\), so the magnitude is \(\|\vec{v}\| = \sqrt{0^2 + (-10)^2} = \sqrt{100} = 10\).

Step 3: Determine the Angle with respect to the Positive X-axis

To find the angle \(\theta\), we use the unit circle understanding with directionality. Since we have no horizontal component and because the vector is purely vertical pointing downwards, \(\theta\) falls directly in the negative y-direction. This corresponds to \(270^{\circ}\) because the angle is measured clockwise from the positive x-axis.

Step 4: Verify the Angle Using the Vector Expression

The vector expression \(\vec{v} = \|\vec{v}\|\langle\cos(\theta), \sin(\theta)\rangle\) suggests that \(\cos(\theta) = 0\) and \(\sin(\theta) = -1\). Thus, \(\theta = 270^{\circ}\) satisfies the condition, confirming the calculated angle is correct.

Key Concepts

Vector Magnitude

Vector magnitude measures how long or large a vector is. Imagine a vector as an arrow pointing in a certain direction; its magnitude is the arrow's length. To calculate the magnitude of any vector \(\vec{v} = (x, y)\), use the formula \(\|\vec{v}\| = \sqrt{x^2 + y^2}\). This formula is essentially the Pythagorean theorem applied to vectors because it calculates the hypotenuse of a right triangle with legs \(x\) and \(y\). To find the magnitude of our specific vector \(\vec{v} = -10 \hat{\jmath}\), we observe there’s no horizontal component, meaning \(x = 0\) and only a vertical component of \(y = -10\). Plug these into the formula to obtain \(\|\vec{v}\| = \sqrt{0^2 + (-10)^2} = 10\). The vector's length is thus 10 units.

Angle Calculation

Calculating an angle \(\theta\) helps understand the vector's direction relative to the positive x-axis. This angle determines where the vector is pointing on a 360-degree circle. For any vector \(\vec{v} = (x, y)\), the angle \(\theta\) can be determined using trigonometric ratios.

• Use \(\tan(\theta) = \frac{y}{x}\). However, this only works when \(x\) is not zero.

Since our vector \(\vec{v} = -10 \hat{\jmath}\) has no horizontal component (i.e., \(x = 0\)), the angle is obtained differently. You need to recognize vectors along the axes:

• For vectors pointing downward, such as \(\vec{v}\), this aligns with the negative y-axis. This means \(\theta = 270^\circ\), completed by the unit circle conventions, measuring clockwise from the positive x-axis.

Unit Circle

The unit circle is a crucial concept in trigonometry for understanding angles and directions. It's a circle centered at the origin (0,0) with a radius of one.

• Its main innovation is linking the circle's points with angles, in radians or degrees, from the positive x-axis.

For any angle \(\theta\):

• \(\cos(\theta)\) represents the x-coordinate.
• \(\sin(\theta)\) represents the y-coordinate.

Consider the vector \(\vec{v} = -10 \hat{\jmath}\): the y-coordinate indicates a downward direction. Recalling the unit circle, below the origin on the vertical axis is \(270^\circ\) where \(\sin(270^\circ) = -1\) and \(\cos(270^\circ) = 0\). This information adds up confirming our angle determination.

Vector Components

Vector components split a vector into the x (horizontal) and y (vertical) values, vital for visualizing and calculating vector properties like magnitude and direction. A vector \(\vec{v} = (x, y)\) is composed of these two parts, describing its placement on a coordinate plane.

• Horizontal component (x): corresponds to left-right movement.
• Vertical component (y): aligns with up-down displacement.

For the specific vector \(\vec{v} = -10 \hat{\jmath}\), notice its vertical-only nature:

• Horizontal (x): 0, meaning there's no side-to-side component.
• Vertical (y): -10, indicating a downward shift.

Comprehending vector components lets you fully understand a vector's influence and its resulting actions in physical space, particularly when considering calculations related to force, motion, and more.