Question

Points \(P\) and \(Q\) are given. Write the vector \(\overrightarrow{P Q}\) in component form and using the standard unit vectors. \(P=(3,2), \quad Q=(7,-2)\)

Short answer

The vector \(\overrightarrow{PQ}\) is \((4, -4)\) or \(4\mathbf{i} - 4\mathbf{j}\).

Step-by-step solution

Identify the Components of Points

First, we need to determine the x and y coordinates of points \(P\) and \(Q\). The coordinates of point \(P\) are \((3, 2)\), while the coordinates of point \(Q\) are \((7, -2)\).

Determine the Components of the Vector

The vector \(\overrightarrow{PQ}\) can be found by subtracting the coordinates of point \(P\) from point \(Q\). The component form of the vector is computed as follows:\[\overrightarrow{PQ} = (x_2 - x_1, y_2 - y_1) = (7 - 3, -2 - 2).\]

Calculate the Components

Now calculate the differences from the previous step:\[x_2 - x_1 = 7 - 3 = 4,\]\[y_2 - y_1 = -2 - 2 = -4.\]Thus, the vector \(\overrightarrow{PQ}\) in component form is \((4, -4)\).

Express in Terms of Standard Unit Vectors

The standard unit vectors in 2D are \(\mathbf{i}\) for the x-direction and \(\mathbf{j}\) for the y-direction. Thus, \(\overrightarrow{PQ}\) can be expressed using these unit vectors as:\[\overrightarrow{PQ} = 4\mathbf{i} - 4\mathbf{j}.\]

Key concepts

Component Form of a Vector

Vectors can be described in component form using their x and y coordinates. This form effectively breaks down a vector into its horizontal (x-axis) and vertical (y-axis) components. To transform points into a vector, you subtract the coordinates of one point from another. Using the component form makes it easy to add and subtract vectors and execute operations like scaling. If we have two points, say \( P = (x_1, y_1) \) and \( Q = (x_2, y_2) \), the vector \( \overrightarrow{PQ} \) can be calculated as \((x_2 - x_1, y_2 - y_1)\).

For example, given points \( P = (3, 2) \) and \( Q = (7, -2) \), the vector \( \overrightarrow{PQ} \) becomes the difference \((7-3, -2-2) = (4, -4)\). Each component of this vector represents the change in the respective axis, with "4" signifying a rightward movement along the x-axis, and "-4" for a downward movement along the y-axis.

Grasping this approach helps solve vector problems since it simplifies understanding of direction and magnitude.

Understanding Unit Vectors

Unit vectors are fundamental in representing vectors because they provide a way to express direction in standard terms. A unit vector has a length of 1, and in two-dimensional space, we commonly use \( \mathbf{i} \) and \( \mathbf{j} \) to denote unit vectors in the x and y directions respectively.

Any vector can be expressed as a combination of these two unit vectors. For a vector \( \overrightarrow{PQ} = (a, b) \), it can be written in terms of unit vectors as \( a\mathbf{i} + b\mathbf{j} \). This notation leverages unit vectors to depict how far and in what direction the vector stretches along each axis.

For the vector \( \overrightarrow{PQ} = (4, -4) \), expressing it with unit vectors gives us \( 4\mathbf{i} - 4\mathbf{j} \). This format helps in visualizing vector paths and is especially useful in physics and engineering applications where directions are crucial.

Vector Subtraction

Vector subtraction is performed by subtracting the corresponding components of two vectors. When dealing with points to find a vector, the head minus tail rule applies: for two points \( P \) and \( Q \), the vector \( \overrightarrow{PQ} \) is found by subtracting the components of point \( P \) from \( Q \).

This process is essential, especially when determining relative positions or changes in position. In mathematical notation, this is expressed as \( (x_2 - x_1, y_2 - y_1) \) — the result gives the vector that points from \( P \) to \( Q \).

• It embraces the rule: \( \overrightarrow{AB} = B - A \).
• It facilitates computing the direction and magnitude of displacement or velocity vectors in physical scenarios.

In our example with points \( P = (3, 2) \) and \( Q = (7, -2) \), subtracting \( P \) from \( Q \) yields the resulting vector components \( (4, -4) \). This operation is straightforward yet instrumental for solving many analytical geometry problems.