Problem 3 - Precalculus with Calculus Previews

Question

Sketch the given vector. Find the magnitude and the smallest positive direction angle of each vector. $$ \langle 5,0\rangle $$

Step-by-Step Solution

Answer

Magnitude is 5, direction angle is 0 degrees.

Step 1: Sketch the Vector

To sketch the vector \( \langle 5, 0 \rangle \), imagine a graph with an x-axis and a y-axis. This vector moves 5 units in the positive x-direction and 0 units in the y-direction. Therefore, you would start at the origin (0,0) and draw an arrow 5 units to the right along the x-axis.

Step 2: Find the Magnitude of the Vector

The magnitude of the vector \( \langle 5, 0 \rangle \) can be calculated using the formula \( \sqrt{x^2 + y^2} \). Plugging in the values, we get \( \sqrt{5^2 + 0^2} = \sqrt{25} = 5 \). Therefore, the magnitude of the vector is 5 units.

Step 3: Find the Smallest Positive Direction Angle

To find the direction angle \( \theta \) of the vector \( \langle 5, 0 \rangle \), we note that it lies along the positive x-axis. The direction angle for a vector on the x-axis in standard position is 0 degrees. Hence, the smallest positive direction angle of this vector is 0 degrees.

Key Concepts

Magnitude of a Vector

The magnitude of a vector is essentially its length. It tells us how far it stretches in the coordinate plane. Calculating the magnitude is straightforward. Use the formula \( \sqrt{x^2 + y^2} \), where \( x \) and \( y \) are the vector's components. For example, if you have a vector \( \langle 5, 0 \rangle \), substitute these values into the formula. This results in \( \sqrt{5^2 + 0^2} = \sqrt{25} = 5 \). Thus, the magnitude is 5 units.
The magnitude is always a non-negative number, as it's a measure of distance. Understanding magnitude helps in visualizing the vector's reach and how far it extends.Consider:

• It doesn't show direction, just how long the vector is.
• It's like measuring the distance between two points.

Direction Angle of a Vector

The direction angle of a vector shows us where the vector points in relation to the horizontal axis. For vectors like \( \langle 5, 0 \rangle \) that align perfectly with the x-axis, determining the direction angle is quite simple. This vector runs straight along the positive x-axis. Therefore, its direction angle is 0 degrees. But what if the vector had different components? To find the direction angle \( \theta \) for any vector, you can use the arctangent function: \[ \theta = \tan^{-1} \left( \frac{y}{x} \right) \]
Direction angles help us understand:

• The tilt or rotation of the vector relative to the x-axis.
• How to navigate using vectors in real-world scenarios, like setting a direction for movement.

In vector analysis, direction angles give clarity about the orientation, making navigation and precise placement more manageable.

Coordinate System

A coordinate system serves as a framework for positioning vectors in space. The most common example is the Cartesian coordinate system, which consists of two axes: x (horizontal) and y (vertical). Points are expressed in pairs, like \( (x, y) \), indicating their position on the plane.With vectors, the coordinate system allows us to visually and mathematically represent their direction and magnitude. A vector like \( \langle 5, 0 \rangle \) shows it moves 5 units along the x-axis, with no movement up or down.Key aspects of coordinate systems include:

• Provides a clear reference point, starting usually from the origin (0,0).
• Helps in visualizing vector addition, subtraction, and understanding relative positions.

When working with vectors, the standard coordinate system helps individuals grasp how vectors function as arrows in space, showing both direction and length, and how different vectors interact or oppose each other.