Problem 19 - Algebra and Trigonometry

Question

In Exercises \(13-20\), let ve the vector from initial point \(P_{1}\) to terminal point \(P_{2}\). Write \(\mathbf{v}\) in terms of \(\mathbf{i}\) and \(\mathbf{j}\) $$P_{1}=(-3,4), P_{2}=(6,4)$$

Step-by-Step Solution

Answer

The vector \(\mathbf{v}\) in terms of \(\mathbf{i}\) and \(\mathbf{j}\) is \(9\mathbf{i} + 0\mathbf{j}\).

Step 1: Understand vector representation

In a 2D plane, each vector \(\mathbf{v}\) can be represented as \(v_x\mathbf{i} + v_y\mathbf{j}\), where \(v_x\) is the x coordinate and \(v_y\) is the y coordinate of the vector \(\mathbf{v}\).

Step 2: Understand how to calculate the coordinates

The coordinates of the vector from point A to B can be calculated as \(B - A\). In this case, \(P_{2}\) is the terminal point and \(\P_{1}\) is the initial point, so we subtract the coordinates of \(\P_{1}\) from \(\P_{2}\). Therefore, the vector \(\mathbf{v}\) can be calculated as \((6-(-3), 4-4)\).

Step 3: Calculate the coordinates

Subtracting the coordinates of \(\P_{1}\) from \(\P_{2}\), we get the coordinates of the vector \(\mathbf{v}\) as \((9, 0)\).

Step 4: Write the vector in terms of i and j

Since the coordinates of the vector \(\mathbf{v}\) are (9, 0), \(\mathbf{v}\) can be written as \(9\mathbf{i} + 0\mathbf{j}\).

Key Concepts

Coordinate System

Understanding a coordinate system is essential in solving problems involving vectors, as it provides a framework to represent positions and directions in a space. The most commonly used system in vector representation is the Cartesian coordinate system. It consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Each point in this two-dimensional space is identified by a pair of numerical coordinates, which are the distances to the point from the two intersecting axes.

For example, the point \( P_{1} = (-3, 4) \) means it's positioned 3 units to the left and 4 units up from the origin, which is the point \( (0, 0) \) where both axes cross. It helps to visualize this as plotting a point on a graph, with \( P_{1} \) as the starting or initial point. When dealing with vector problems, the coordinate points determine the vector's magnitude and direction.

Vector Subtraction

Vector subtraction is a fundamental operation in vector algebra, allowing us to find the difference between two points or vectors. When you subtract one vector from another, you effectively measure the change from one point to another. The formula for vector subtraction is quite straightforward—subtracts each component of the vector separately.

As the example \(P_{1}=(-3,4) \) and \(P_{2}=(6,4)\) indicates, we calculate the resultant vector \(\mathbf{v}\) as \(P_{2} - P_{1}\). This gives us the components of the new vector in the Cartesian plane. By subtracting, we find the horizontal and vertical distances between the two points, which collectively define the new vector's direction and length.

Unit Vectors

Unit vectors play a critical role in vector representation because they provide a standard for expressing vectors. A unit vector has a magnitude (or length) of exactly one unit and it indicates direction only. The standard unit vectors in a two-dimensional Cartesian coordinate system are denoted by \( \mathbf{i} \) and \( \mathbf{j} \).

The \( \mathbf{i} \) vector represents one unit of length in the direction of the x-axis, while \( \mathbf{j} \) represents one unit of length in the direction of the y-axis. When expressing any other vector, like \( \mathbf{v} \) in the example given, we use multiples of these unit vectors to describe its direction and magnitude in the x and y directions. As shown, \( \mathbf{v} = 9\mathbf{i} + 0\mathbf{j} \) simply indicates that the vector \( \mathbf{v} \) has a component of 9 units in the direction of the x-axis and none in the direction of the y-axis.