Problem 73 - Trigonometry

Question

These questions are available for instructors to help assess if you have successfully met the learning objectives for this section. Draw \(\mathbf{V}=\langle 2,3\rangle\) in standard position.

Step-by-Step Solution

Answer

Draw a line from (0,0) to (2,3) with an arrowhead at (2,3).

Step 1: Understanding Vectors in Standard Position

A vector in standard position starts at the origin, point \(0,0\), and ends at a point determined by its coordinates. The vector \(\mathbf{V}=\langle 2,3\rangle\) means it starts at \(0,0\) and ends at \(2,3\).

Step 2: Plotting the Origin

Begin your drawing by plotting the origin point, which is \(0,0\). This is the starting point of our vector in standard position.

Step 3: Plotting the Endpoint

Next, plot the endpoint of the vector, which is given by its coordinates. For \(\mathbf{V}=\langle 2,3\rangle\), plot the point \(2,3\) on the graph.

Step 4: Drawing the Vector

Draw an arrowed line from the origin point \(0,0\) to the endpoint \(2,3\). Ensure the arrowhead is at \(2,3\) indicating the direction of the vector.

Key Concepts

Vectors in Standard Position

Vectors are mathematical entities that have both magnitude and direction. When dealing with vectors in standard position, they specifically start at the origin of a coordinate plane, at the point \( (0, 0) \). This is a convention that simplifies the understanding and manipulation of vectors in math and physics.

For example, when we talk about the vector \( \mathbf{V} = \langle 2,3 \rangle \), it's crucial to start plotting from the origin. This essentially means that the vector 'begins' at \( (0, 0) \) and 'ends' at another point specified by its components, in this case at \( (2,3) \). The numbers \( 2 \) and \( 3 \) represent the movement from the origin along the x-axis and y-axis, respectively.

• Starting point: Origin \( (0, 0) \)
• Ending point: Determined by vector components \( (2,3) \)

By understanding vectors in standard position, it becomes easier to perform operations such as vector addition, subtraction, and to apply these concepts in both geometric and physical contexts.

Graphing Vectors

Graphing vectors involves visually representing the vector on a coordinate plane. This process requires knowing both the starting and ending points of the vector. In standard position, the starting point is always the origin \( (0, 0) \).

To graph the vector \( \mathbf{V} = \langle 2,3 \rangle \), start by plotting the origin. From there, use the components of the vector to determine the endpoint. In this case, you move 2 units to the right along the x-axis and 3 units up along the y-axis, arriving at point \( (2, 3) \).

• Begin at the origin \( (0, 0) \)
• Move horizontally based on the x-component
• Move vertically based on the y-component
• Draw an arrow from the start to the end point

The arrow demonstrates both the direction and magnitude of the vector. This visual representation helps in better understanding the relationships between multiple vectors and can aid in solving more complex problems involving vector operations.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that allows geometric problems to be solved using algebra and a coordinate system. It involves plotting points, lines, and curves on a plane defined by a coordinate system, typically the Cartesian plane.

Vectors are an integral part of coordinate geometry because they can represent lines, directions, and displacements on the plane. For instance, the vector \( \mathbf{V} = \langle 2,3 \rangle \) can be plotted on the Cartesian plane to illustrate movement from the origin to the point \( (2, 3) \).

• Utilizes the Cartesian coordinate system
• Solves geometrical problems using algebra
• Represents vectors as directed line segments

By mastering coordinate geometry, one can easily transition between geometrical concepts and algebraic expressions, making it a powerful tool in both pure mathematics and practical applications such as engineering and physics.