Question

Vectors \(\vec{x}\) and \(\vec{y}\) are given. Sketch \(\vec{x}, \vec{y}, \vec{x}+\vec{y},\) and \(\vec{x}-\vec{y}\) on the same Cartesian axes. $$ \vec{x}=\left[\begin{array}{l} 3 \\ 1 \end{array}\right], \vec{y}=\left[\begin{array}{c} 1 \\ -2 \end{array}\right] $$

Short answer

Vectors are \( \vec{x} = \begin{bmatrix} 3 \\ 1 \end{bmatrix} \), \( \vec{y} = \begin{bmatrix} 1 \\ -2 \end{bmatrix} \), \( \vec{x} + \vec{y} = \begin{bmatrix} 4 \\ -1 \end{bmatrix} \), \( \vec{x} - \vec{y} = \begin{bmatrix} 2 \\ 3 \end{bmatrix} \).

Step-by-step solution

Understand the Given Vectors

We are given two vectors, \( \vec{x} = \begin{bmatrix} 3 \ 1 \end{bmatrix} \) and \( \vec{y} = \begin{bmatrix} 1 \ -2 \end{bmatrix} \). The vector \( \vec{x} \) represents a point at (3, 1) on the Cartesian plane, and the vector \( \vec{y} \) represents a point at (1, -2).

Calculate the Sum of Vectors

To find the vector \( \vec{x} + \vec{y} \), we add the corresponding components of the vectors: \( \vec{x} + \vec{y} = \begin{bmatrix} 3 \ 1 \end{bmatrix} + \begin{bmatrix} 1 \ -2 \end{bmatrix} = \begin{bmatrix} 3 + 1 \ 1 + (-2) \end{bmatrix} = \begin{bmatrix} 4 \ -1 \end{bmatrix} \).

Calculate the Difference of Vectors

To find the vector \( \vec{x} - \vec{y} \), we subtract the corresponding components of the vectors: \( \vec{x} - \vec{y} = \begin{bmatrix} 3 \ 1 \end{bmatrix} - \begin{bmatrix} 1 \ -2 \end{bmatrix} = \begin{bmatrix} 3 - 1 \ 1 - (-2) \end{bmatrix} = \begin{bmatrix} 2 \ 3 \end{bmatrix} \).

Sketch the Vectors on Cartesian Axes

Begin by drawing the x and y axes on paper. Plot the vector \( \vec{x} = \begin{bmatrix} 3 \ 1 \end{bmatrix} \) starting from the origin and pointing to the point (3, 1). Next, plot \( \vec{y} = \begin{bmatrix} 1 \ -2 \end{bmatrix} \) from the origin to (1, -2). Plot \( \vec{x} + \vec{y} = \begin{bmatrix} 4 \ -1 \end{bmatrix} \) as a new vector starting from the origin to (4, -1). Lastly, plot \( \vec{x} - \vec{y} = \begin{bmatrix} 2 \ 3 \end{bmatrix} \) as a vector from the origin to (2, 3).

Key concepts

vectors

Vectors are fundamental components in mathematics and physics, often used to describe quantities possessing both magnitude and direction. Unlike scalars which only have magnitude, vectors tell us how far and in which direction to move.

• For example, a vector can represent a force acting on an object with a particular strength and direction.
• In mathematics, vectors are typically expressed in terms of their components, often using coordinates in a space.

In the given exercise, two-dimensional vectors are used, given by their components along the x and y axes.

• \( \vec{x} = \begin{bmatrix} 3 \ 1 \end{bmatrix} \) indicates a point (or direction) at (3,1) on a plane.
• \( \vec{y} = \begin{bmatrix} 1 \ -2 \end{bmatrix} \) points to (1,-2).

These vectors can visually be represented as arrows on Cartesian coordinates, where the origin is the starting point and the end of the vector shows its terminal point.

Cartesian plane

The Cartesian plane is a two-dimensional space defined by two perpendicular axes: the x-axis and the y-axis. This coordinate system helps us visually understand and plot vectors as well as perform operations on them.

• Each axis represents a different dimension, with the intersection at the origin (0,0).
• Points on the plane are described using ordered pairs \((x, y)\).

When plotting vectors like \(\vec{x}\) and \(\vec{y}\) from the exercise, you can think of starting at the origin for each vector:

• Draw an arrow from (0,0) to (3,1) for \(\vec{x}\).
• Similarly, plot \(\vec{y}\) from (0,0) to (1,-2).

Performing vector operations like addition and subtraction also take place on the Cartesian plane, facilitating easy visualization.

• For vector addition, you can use the "tip-to-tail" method, connecting the tail of one vector to the tip of the other.
• For subtraction, visualize adding a negative of the vector.

vector subtraction

Vector subtraction is the process of finding the vector difference between two vectors. It involves taking one vector and effectively reversing the direction of the vector we are subtracting, then performing an addition.

• Mathematically, subtracting vector \(\vec{b}\) from vector \(\vec{a}\) can be written as \(\vec{a} - \vec{b}\).
• This entails subtracting each component of \(\vec{b}\) from the corresponding component of \(\vec{a}\).

In the provided exercise, we subtract \(\vec{y} = \begin{bmatrix} 1 \ -2 \end{bmatrix}\) from \(\vec{x} = \begin{bmatrix} 3 \ 1 \end{bmatrix}\) like so:

• Calculate \(3 - 1\) for the x-component.
• Calculate \(1 - (-2)\) for the y-component.
• The result is the vector \(\vec{x} - \vec{y} = \begin{bmatrix} 2 \ 3 \end{bmatrix}\).

Visually, on the Cartesian plane, you would notice that by reversing and adding \(\vec{y}\), we land on the components (2,3), which represents the new vector from the origin to this point.