Question

For the given vectors \(\mathbf{u}\) and \(\mathbf{v}\), evaluate the following expressions. a. \(3 \mathbf{u}+2 \mathbf{v}\) b. \(4 \mathbf{u}-\mathbf{v}\) c. \(|\mathbf{u}+3 \mathbf{v}|\) $$\mathbf{u}=\langle-5,0,2\rangle, \mathbf{v}=\langle 3,1,1\rangle$$

Short answer

Question: Determine the vectors resulting from the following operations: a. \(3 \mathbf{u}+2 \mathbf{v}\), b. \(4 \mathbf{u}-\mathbf{v}\), and c. \(|\mathbf{u}+3 \mathbf{v}|\), given that \(\mathbf{u} = \langle-5,0,2\rangle\) and \(\mathbf{v} = \langle 3,1,1\rangle\). Answer: a. \(3 \mathbf{u}+2 \mathbf{v} = \langle-9,2,8\rangle\), b. \(4 \mathbf{u}-\mathbf{v} = \langle-23,-1,7\rangle\), and c. \(|\mathbf{u}+3 \mathbf{v}| = \sqrt{50}\).

Step-by-step solution

a. \(3 \mathbf{u}+2 \mathbf{v}\)

To find \(3 \mathbf{u}+2 \mathbf{v}\), we first need to multiply each vector by their respective scalars (3 and 2) and then add the two resulting vectors. Let's do this: 1. Multiply \(\mathbf{u}\) by 3: $$3 \mathbf{u} = 3 \langle-5,0,2\rangle = \langle-15,0,6\rangle$$ 2. Multiply \(\mathbf{v}\) by 2: $$2 \mathbf{v} = 2 \langle 3,1,1\rangle = \langle6,2,2\rangle$$ 3. Add the two vectors: $$3 \mathbf{u}+2 \mathbf{v} = \langle-15,0,6\rangle + \langle6,2,2\rangle = \langle-9,2,8\rangle$$ So, \(3 \mathbf{u}+2 \mathbf{v} = \langle-9,2,8\rangle\).

b. \(4 \mathbf{u}-\mathbf{v}\)

To find \(4 \mathbf{u}-\mathbf{v}\), we first need to multiply \(\mathbf{u}\) by 4 and then subtract \(\mathbf{v}\) from the resulting vector. Let's do this: 1. Multiply \(\mathbf{u}\) by 4: $$4 \mathbf{u} = 4 \langle-5,0,2\rangle = \langle-20,0,8\rangle$$ 2. Subtract \(\mathbf{v}\): $$4 \mathbf{u}-\mathbf{v} = \langle-20,0,8\rangle - \langle 3,1,1\rangle = \langle-23,-1,7\rangle$$ So, \(4 \mathbf{u}-\mathbf{v} = \langle-23,-1,7\rangle\).

c. \(|\mathbf{u}+3 \mathbf{v}|\)

To find \(|\mathbf{u}+3 \mathbf{v}|\), we first need to compute \(\mathbf{u}+3 \mathbf{v}\), and then find the magnitude of the resulting vector. Let's do this: 1. Multiply \(\mathbf{v}\) by 3: $$3 \mathbf{v} = 3 \langle 3,1,1\rangle = \langle9,3,3\rangle$$ 2. Add \(\mathbf{u}\): $$\mathbf{u}+3 \mathbf{v} = \langle-5,0,2\rangle + \langle9,3,3\rangle = \langle4,3,5\rangle$$ 3. Find the magnitude of the resulting vector: $$|\mathbf{u}+3 \mathbf{v}| = |\langle4,3,5\rangle| = \sqrt{4^2+3^2+5^2} = \sqrt{50}$$ So, \(|\mathbf{u}+3 \mathbf{v}| = \sqrt{50}\).

Key concepts

Vector Addition

Vector addition occurs when you combine two or more vectors together to create a new vector. Imagine vectors as arrows, each with a direction and magnitude. When you add vectors, you simply adjust their directions and lengths, creating a resulting vector that represents the combined effect.

To add vectors, you sum their corresponding components. Suppose you have two vectors:

• \( \mathbf{a} = \langle a_1, a_2, a_3 \rangle \)
• \( \mathbf{b} = \langle b_1, b_2, b_3 \rangle \)

The sum of these vectors would be: \[\mathbf{a} + \mathbf{b} = \langle a_1 + b_1, a_2 + b_2, a_3 + b_3 \rangle\]This method of adding vectors allows for easy combination of forces or movements in physics or engineering. It is important to stay organized and careful with signs (positive or negative), especially when combining directions which can cancel each other out if they go opposite directions.

Scalar Multiplication

Scalar multiplication is the process of stretching or shrinking a vector by multiplying it with a scalar, a single number. This operation changes the magnitude (length) of the vector but not its direction.

Suppose you have a vector \( \mathbf{c} = \langle c_1, c_2, c_3 \rangle \) and want to multiply it by a scalar \( k \). The resulting vector is calculated by multiplying each component by \( k \): \[k \cdot \mathbf{c} = \langle k \cdot c_1, k \cdot c_2, k \cdot c_3 \rangle\]

• If \( k \) is greater than 1, the vector is elongated.
• If \( k \) is between 0 and 1, the vector is shortened.
• If \( k \) is negative, the vector changes direction.

This operation is particularly useful in graphics and physics, where adjusting the size of vectors can model various scenarios, like scaling up a graphic or changing velocity.

Vector Magnitude

The magnitude of a vector denotes its length, reflecting how much of a quantity is being described. It is a crucial concept when analyzing vectors as you often need to know the strength or extent of what the vector represents.

To find the magnitude of a vector \( \mathbf{d} = \langle d_1, d_2, d_3 \rangle \), you use the formula: \[|\mathbf{d}| = \sqrt{d_1^2 + d_2^2 + d_3^2}\]This formula is derived from the Pythagorean theorem, which applies to right-angled triangles. In a three-dimensional space, it helps find the diagonal distance between the origin and the vector's endpoint by summing the squares of each component and taking the square root.

Knowing the magnitude is essential when working with real-world applications where distances, forces, and other measurements matter. With this knowledge, you can scale, compare, and utilize vectors efficiently in calculations and designs.