Question

Points \(P\) and \(Q\) are given. Write the vector \(\overrightarrow{P Q}\) in component form and using the standard unit vectors. \(P=(0,3,-1), \quad Q=(6,2,5)\)

Short answer

\(\overrightarrow{PQ} = 6\mathbf{i} - \mathbf{j} + 6\mathbf{k}\).

Step-by-step solution

Find the Component Form

To find the vector \(\overrightarrow{PQ}\), we subtract the coordinates of point \(P\) from the coordinates of point \(Q\). The formula for the component form of the vector \(\overrightarrow{PQ}\) is: \( (x_2 - x_1, y_2 - y_1, z_2 - z_1) \).Here, \(x_1 = 0\), \(y_1 = 3\), \(z_1 = -1\) and \(x_2 = 6\), \(y_2 = 2\), \(z_2 = 5\).Substituting these values, we have:\(\overrightarrow{PQ} = (6 - 0, 2 - 3, 5 + 1) = (6, -1, 6)\).

Express Using Standard Unit Vectors

The component form of vector \(\overrightarrow{PQ}\) can be written using the standard unit vectors \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\).Standard unit vectors are \(\mathbf{i} = (1,0,0)\), \(\mathbf{j} = (0,1,0)\), \(\mathbf{k} = (0,0,1)\).Thus, the vector \(\overrightarrow{PQ} = (6, -1, 6)\) can be expressed as:\[\overrightarrow{PQ} = 6\mathbf{i} - \mathbf{j} + 6\mathbf{k}\].

Key concepts

Component Form

When we talk about the component form of a vector, we're really discussing how to express a vector in a way that highlights the change in each coordinate direction. Imagine you have two points, like our points \( P = (0,3,-1) \) and \( Q = (6,2,5) \). The vector \( \overrightarrow{PQ} \) represents the direction and magnitude from point \( P \) to point \( Q \).

To find the component form, we subtract the coordinates of point \( P \) from point \( Q \). The general formula is \( (x_2 - x_1, y_2 - y_1, z_2 - z_1) \). This gives us a new set of components that define the vector.

• For the \( x \) component: \( 6 - 0 = 6 \)
• For the \( y \) component: \( 2 - 3 = -1 \)
• For the \( z \) component: \( 5 - (-1) = 6 \)

So, the vector \( \overrightarrow{PQ} \) is \( (6, -1, 6) \). This component form tells us exactly how much we need to move in the \( x \), \( y \), and \( z \) directions to go from point \( P \) to point \( Q \). It essentially breaks down the vector into manageable pieces along each axis.

Unit Vectors

Unit vectors are vital in vector calculus as they provide a way to express vectors in a standardized form. A unit vector has a magnitude of 1 and indicates direction along an axis without stretching or shrinking the vector itself.

• \( \mathbf{i} \) stands for the unit vector in the direction of the x-axis: \( (1, 0, 0) \).
• \( \mathbf{j} \) represents the unit vector in the direction of the y-axis: \( (0, 1, 0) \).
• \( \mathbf{k} \) is the unit vector in the direction of the z-axis: \( (0, 0, 1) \).

Whenever you express a vector like \( \overrightarrow{PQ} = (6, -1, 6) \) using unit vectors, you're effectively expanding it using a combination of these three directions.

Let's see how it works: \[ \overrightarrow{PQ} = 6\mathbf{i} - \mathbf{j} + 6\mathbf{k} \]In this equation, the coefficients 6, -1, and 6 scale the unit vectors to create a composite vector that mimics the original movement from point \( P \) to point \( Q \). This notation is incredibly useful in physics and engineering because it helps visualize vector direction and contribution relative to each axis.

Vector Subtraction

Vector subtraction is the key operation we use to find the difference between two points in space, effectively creating another vector. By subtracting, we determine how far and in what direction these points are from each other.

Let's break it down using our example of points \( P \) and \( Q \). To get from \( P \) to \( Q \), we subtract the coordinates of \( P \) from \( Q \). This is where vector subtraction shines:

• The change in the \( x \) direction: \( 6 - 0 = 6 \)
• The change in the \( y \) direction: \( 2 - 3 = -1 \)
• The change in the \( z \) direction: \( 5 - (-1) = 6 \)

Each of these results forms the component of a new vector pointing from \( P \) to \( Q \), hence \( \overrightarrow{PQ} = (6, -1, 6) \). Subtracting in this way combines both magnitude and direction to accurately describe the displacement between two points. This concept is crucial in navigation and physics, where knowing how to adjust from one position to another is essential.