Question

Let \(\mathbf{u}=\langle 3,-4\rangle, \mathbf{v}=\langle 1,1\rangle,\) and \(\mathbf{w}=\langle 1,0\rangle\) Find two vectors parallel to \(\mathbf{v}\) with three times the magnitude of \(\mathbf{v}\)

Short answer

Question: Find two vectors parallel to vector \(\mathbf{v} = \langle 1, 1 \rangle\) with three times the magnitude of \(\mathbf{v}\). Answer: The two vectors parallel to \(\mathbf{v}\) with three times the magnitude of \(\mathbf{v}\) are \(\mathbf{a} = \langle 3, 3 \rangle\) and \(\mathbf{b} = \langle -3, -3 \rangle\).

Step-by-step solution

Find the magnitude of \(\mathbf{v}\)

To find the magnitude of \(\mathbf{v}\), we use the formula \(\|\mathbf{v}\| = \sqrt{v_x^2 + v_y^2}\) where \(v_x\) and \(v_y\) are the respective x and y components of \(\mathbf{v}\). In this case, \(v_x = 1\) and \(v_y = 1\). So, the magnitude is given by: \(\|\mathbf{v}\| = \sqrt{1^2 + 1^2} = \sqrt{2}\)

Perform scalar multiplication

Now, we need to find two vectors parallel to \(\mathbf{v}\) with three times the magnitude of \(\mathbf{v}\). Let's call these vectors \(\mathbf{a}\) and \(\mathbf{b}\). To make a vector parallel to \(\mathbf{v}\), we need to multiply \(\mathbf{v}\) by a scalar. In this case, we will multiply by 3 and -3, to obtain two parallel vectors: \(\mathbf{a} = 3\mathbf{v} = 3 \langle 1,1\rangle = \langle 3,3 \rangle\) \(\mathbf{b} = -3\mathbf{v} = -3 \langle 1,1\rangle = \langle -3,-3 \rangle\)

Verify the magnitudes

Now, let's verify if these vectors have three times the magnitude of \(\mathbf{v}\): \(\|\mathbf{a}\| = \sqrt{3^2 + 3^2} = \sqrt{18} = 3\sqrt{2}\), which is 3 times the magnitude of \(\mathbf{v}\) \(\|\mathbf{b}\| = \sqrt{(-3)^2 + (-3)^2} = \sqrt{18} = 3\sqrt{2}\), which is also 3 times the magnitude of \(\mathbf{v}\) Thus, we found two vectors parallel to \(\mathbf{v}\) with three times the magnitude of \(\mathbf{v}\): \(\mathbf{a} = \langle 3,3 \rangle\) and \(\mathbf{b} = \langle -3,-3 \rangle\).

Key concepts

Vector Magnitude

The magnitude of a vector, often denoted as \( \|\mathbf{v}\| \), can be thought of as the length of the vector. It provides the size of the vector irrespective of its direction. To calculate the magnitude of a vector in a Cartesian coordinate system, we use the Pythagorean theorem. For a vector \( \mathbf{v} = \langle v_x, v_y \rangle \), the magnitude is found using the formula: \( \|\mathbf{v}\| = \sqrt{v_x^2 + v_y^2} \).
This formula essentially extends the concept of distance from the origin to the point \( \langle v_x, v_y \rangle \) in a two-dimensional space. Thus, for the given vector \( \mathbf{v} = \langle 1,1 \rangle \), we calculate its magnitude:

• Square each component: \( 1^2 + 1^2 = 2 \).
• Take the square root of the sum: \( \sqrt{2} \).

This value \( \sqrt{2} \) signifies the length of the vector \( \mathbf{v} \) in the plane. The concept of vector magnitude is essential when comparing vectors or when performing operations that involve scaling, such as scalar multiplication, or when finding vectors that are parallel but with different magnitudes.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar (a single number). This operation scales the vector, altering its magnitude without changing its direction. If a vector \( \mathbf{v} = \langle v_x, v_y \rangle \) is multiplied by a scalar \( k \), the resulting vector \( k\mathbf{v} \) is given by \( \langle k v_x, k v_y \rangle \).
The direction remains the same if \( k \) is positive and is reversed if \( k \) is negative. This operation is incredibly useful as it allows us to manipulate the vector's magnitude while retaining, or intentionally reversing, its orientation.
For example, to find vectors \( \mathbf{a} \) and \( \mathbf{b} \) that are parallel to \( \mathbf{v} = \langle 1, 1 \rangle \) with three times its magnitude, we perform the following:

• Multiply by \( 3 \): \( 3 \langle 1, 1 \rangle = \langle 3, 3 \rangle \).
• Multiply by \( -3 \): \( -3 \langle 1, 1 \rangle = \langle -3, -3 \rangle \).

Both vectors are parallel to \( \mathbf{v} \), with \( \mathbf{a} \) pointing in the same direction and \( \mathbf{b} \) in the opposite direction, each possessing a magnitude \( 3\sqrt{2} \), exactly three times that of \( \mathbf{v} \).

Parallel Vectors

Parallel vectors share the same or opposite direction, meaning one is a scalar multiple of the other. Two vectors \( \mathbf{u} \) and \( \mathbf{v} \) are considered parallel if there exists a scalar \( k \) such that \( \mathbf{u} = k\mathbf{v} \). This relationship confirms that the vectors have identical or binary opposite orientations, despite possibly differing in length.
For example, given \( \mathbf{v} = \langle 1, 1 \rangle \), a vector \( \mathbf{a} = \langle 3, 3 \rangle \) is parallel because it can be produced by multiplying \( \mathbf{v} \) by \( 3 \). Similarly, \( \mathbf{b} = \langle -3, -3 \rangle \) is parallel to \( \mathbf{v} \) because it results from multiplying \( \mathbf{v} \) by \( -3 \).
Parallel vectors are central in vector calculus and physics, as they help describe directions, velocities, and forces along the same line of action. Recognizing that vectors such as these are parallel regardless of how far they are scaled can be critical when solving problems related to vector projections and equilibria. Understanding these properties is crucial for accurately modeling physical phenomena and solving geometric problems using vectors.