Question

Let \(\mathbf{u}=\langle 3,-4\rangle, \mathbf{v}=\langle 1,1\rangle,\) and \(\mathbf{w}=\langle-1,0\rangle .\) Carry out the following computations. Which has the greater magnitude, \(\mathbf{u}-\mathbf{v}\) or \(\mathbf{w}-\mathbf{u} ?\)

Short answer

Given vectors \(\mathbf{u}=\langle 3, -4\rangle\), \(\mathbf{v}=\langle 1, 1\rangle\), and \(\mathbf{w}=\langle -1, 0\rangle\). Answer: The vector \(\mathbf{w}-\mathbf{u}\) has a greater magnitude than the vector \(\mathbf{u}-\mathbf{v}\).

Step-by-step solution

Subtract the Vectors

We first need to calculate the vectors \(\mathbf{u}-\mathbf{v}\) and \(\mathbf{w}-\mathbf{u}\). To do this, subtract the corresponding components of the vectors. For \(\mathbf{u}-\mathbf{v}\): \[(3 - 1, -4 - 1) = (2, -5)\] For \(\mathbf{w}-\mathbf{u}\): \[(-1 - 3, 0 - (-4)) = (-4, 4)\] So, \(\mathbf{u}-\mathbf{v}=\langle 2, -5\rangle\) and \(\mathbf{w}-\mathbf{u}=\langle-4, 4\rangle\).

Calculate the Magnitudes

Next, we need to calculate the magnitudes of these two vectors. The magnitude of a vector \(\langle x, y \rangle\) is given by the formula \(\sqrt{x^2 + y^2}\). For the vector \(\mathbf{u}-\mathbf{v}=\langle 2, -5\rangle\): \[||\mathbf{u}-\mathbf{v}|| = \sqrt{2^2 + (-5)^2} = \sqrt{4 + 25} = \sqrt{29}\] For the vector \(\mathbf{w}-\mathbf{u}=\langle -4, 4\rangle\): \[||\mathbf{w}-\mathbf{u}|| = \sqrt{(-4)^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32}\]

Compare the Magnitudes

Now, we compare the magnitudes of the vectors \(\mathbf{u}-\mathbf{v}\) and \(\mathbf{w}-\mathbf{u}\). Since \(\sqrt{29} \approx 5.39\) and \(\sqrt{32} \approx 5.66\), we see that: \[||\mathbf{w}-\mathbf{u}|| > ||\mathbf{u}-\mathbf{v}||\] Thus, the vector \(\mathbf{w}-\mathbf{u}\) has a greater magnitude than the vector \(\mathbf{u}-\mathbf{v}\).

Key concepts

Vector Subtraction

Vector subtraction is a fundamental concept in vector algebra that helps us determine the difference between two vectors by subtracting their corresponding components.
It's similar to subtracting numbers, but here we deal with pairs of numbers called components.

• Each vector is represented as an ordered pair or triplet, depending on the dimensions.
• To subtract vectors, subtract the corresponding components.

Given vectors \(\mathbf{u} = \langle 3, -4 \rangle\) and \(\mathbf{v} = \langle 1, 1 \rangle\), subtracting \(\mathbf{v}\) from \(\mathbf{u}\) means:
\[\mathbf{u} - \mathbf{v} = \langle 3 - 1, -4 - 1 \rangle = \langle 2, -5 \rangle\]
Now, for \(\mathbf{w} = \langle -1, 0 \rangle\) and subtracting \(\mathbf{u}\), the subtraction \( \mathbf{w} - \mathbf{u} \) is:
\[\mathbf{w} - \mathbf{u} = \langle -1 - 3, 0 - (-4) \rangle = \langle -4, 4 \rangle\] This calculation forms the basis for further analysis, such as finding magnitudes or distances between points.

Euclidean Distance

The Euclidean distance is a measure that shows the straight-line distance between two points in Euclidean space.
Often used in geometry, it's calculated using the Pythagorean theorem.
When discussing vectors, the distance can translate to a vector's length or magnitude:

• In 2D space, if a vector \( \mathbf{v} = \langle x, y \rangle \), its Euclidean distance or magnitude from the origin is \( \sqrt{x^2 + y^2} \).
• For vectors \( \mathbf{u} - \mathbf{v} \) and \( \mathbf{w} - \mathbf{u} \), we find this distance by calculating the magnitude.

For instance, for \( \mathbf{u} - \mathbf{v} = \langle 2, -5 \rangle \):
\[||\mathbf{u} - \mathbf{v}|| = \sqrt{2^2 + (-5)^2} = \sqrt{4 + 25} = \sqrt{29}\]
Similarly, for \( \mathbf{w} - \mathbf{u} = \langle -4, 4 \rangle \):
\[||\mathbf{w} - \mathbf{u}|| = \sqrt{(-4)^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32}\] These calculations help find which vector is longer and therefore has a greater distance from its origin.

Vector Components

Vector components are the individual values that make up a vector.
They determine the direction and magnitude of the vector when combined.
In a 2D vector, each component corresponds to horizontal and vertical distances on a plane:

• A vector in two dimensions is often written as \( \langle x, y \rangle \), where \( x \) is the horizontal component, and \( y \) is the vertical component.
• The components indicate how far and in which direction the vector extends from its origin.

Take the vector \( \langle 2, -5 \rangle \):

• Here, 2 is the horizontal component, moving right (positive direction) on the x-axis.
• -5 is the vertical component, moving down (negative direction) on the y-axis.

Similarly, for the vector \( \langle -4, 4 \rangle \):

• -4 extends left on the x-axis, indicating the negative direction.
• 4 extends upwards on the y-axis, indicating the positive direction.

Understanding these components is crucial for performing vector operations like addition, subtraction, and calculating their magnitude.