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. \(|\mathbf{u}+3 \mathbf{v}|\) $$\mathbf{u}=\langle-2,1,-2\rangle, \mathbf{v}=\langle 1,1,1\rangle$$

Short answer

Question: Evaluate the following expressions for the given vectors \(\mathbf{u} = \langle-2, 1, -2\rangle\) and \(\mathbf{v} = \langle1, 1, 1\rangle\): a. \(3\mathbf{u}+2\mathbf{v}\), b. \(4\mathbf{u}-\mathbf{v}\), c. \(|\mathbf{u}+3\mathbf{v}|\). Answer: a. \(3\mathbf{u}+2\mathbf{v} = \langle-4,5,-4\rangle\) b. \(4\mathbf{u}-\mathbf{v} = \langle-9,3,-9\rangle\) c. \(|\mathbf{u}+3 \mathbf{v}| = \sqrt{18}\)

Step-by-step solution

Scalar Multiplication

Multiply the given scalar with respective vectors as: a. \(3\mathbf{u} = 3\langle-2,1,-2\rangle = \langle-6,3,-6\rangle\) b. \(4\mathbf{u} = 4\langle-2,1,-2\rangle = \langle-8,4,-8\rangle\) c. \(3\mathbf{v} = 3\langle 1,1,1\rangle = \langle3,3,3\rangle\)

Vector Addition and Subtraction

Add/subtract the vectors as required for the expressions: a. \(3\mathbf{u}+2\mathbf{v} = \langle-6,3,-6\rangle + 2\langle 1,1,1\rangle = \langle-6,3,-6\rangle + \langle2,2,2\rangle = \langle-4,5,-4\rangle\) b. \(4\mathbf{u}-\mathbf{v} = \langle-8,4,-8\rangle - \langle 1,1,1\rangle = \langle-9,3,-9\rangle\)

Magnitude Calculation

Add the vectors for expression (c) and then find the magnitude: \(\mathbf{u}+3 \mathbf{v} = \langle-2,1,-2\rangle + \langle3,3,3\rangle = \langle1,4,1\rangle\) Now, calculate the magnitude of the resulting vector: \(|\mathbf{u} + 3\mathbf{v}| = |\langle1,4,1\rangle| = \sqrt{1^2 + 4^2 + 1^2} = \sqrt{18}\). #Conclusion# a. \(3\mathbf{u}+2\mathbf{v} = \langle-4,5,-4\rangle\) b. \(4\mathbf{u}-\mathbf{v} = \langle-9,3,-9\rangle\) c. \(|\mathbf{u}+3 \mathbf{v}| = \sqrt{18}\)

Key concepts

Scalar Multiplication

Scalar multiplication is a fundamental operation involving vectors. Here, we multiply each component of a vector by a scalar. A scalar is just a real number that can stretch or shrink the vector, affecting its magnitude but not its direction (unless you multiply by a negative number, which also inverts its direction). For example, in the exercise, scalar multiplication is used in expressions like:

• For vector \(\mathbf{u} = \langle-2,1,-2\rangle\), multiplying by 3, we get \(3\mathbf{u} = 3\times\langle-2,1,-2\rangle = \langle-6,3,-6\rangle\).

Perform scalar multiplication component-wise:

• Multiply each component of the vector by the scalar.
• This operation can either increase or decrease the magnitude of the vector based on the scalar value.
• Ensure to keep track of the signs, as multiplying by negative values will reverse directions.

The process is simple: if \(v = \langle a, b, c\rangle\) and the scalar is \(k\), then \(k \cdot v = \langle k \cdot a, k \cdot b, k \cdot c\rangle\). Easy, right?

Vector Addition

Vector addition involves combining two or more vectors to produce another vector. It's like finding a middle path between two directions. Each component of the vectors is added together to form the result. This operation can be visualized geometrically by the tip-to-tail method: aligning the end of one vector to the start of another. However, in computation, we focus on the components directly.

• To add two vectors, like \(\mathbf{u} = \langle-6,3,-6\rangle\) and \(2\mathbf{v} = \langle2,2,2\rangle\), you add corresponding components: \(-6+2 \rightarrow -4;\ 3+2 \rightarrow 5;\ -6+2 \rightarrow -4\).
• The resulting vector \(\mathbf{u} + 2\mathbf{v} = \langle-4,5,-4\rangle\).

Subtraction of vectors follows similarly but includes changing the sign of each component of the vector being subtracted. Thus, for \( \mathbf{v} = \langle 1,1,1 \rangle \), subtracting it from \(\mathbf{u} = \langle -8,4,-8 \rangle\) produces \(\mathbf{u} - \mathbf{v} = \langle -9,3,-9 \rangle\). It's straightforward: \(\langle a,b,c\rangle + \langle d,e,f\rangle = \langle a+d, b+e, c+f\rangle\). Vectors help us visualize and compute combined directions or forces!

Vector Magnitude

Vector magnitude measures the length of the vector, reflecting how far away its terminal point is from the origin. To compute it, you use the Pythagorean theorem in three dimensions. In simple terms, it's like calculating the hypotenuse of a triangle in space, given its other sides. For a vector \(\mathbf{r} = \langle a,b,c\rangle\), the magnitude \(|\mathbf{r}|\) is calculated using the formula: \[ |\mathbf{r}| = \sqrt{a^2 + b^2 + c^2} \] This gives a non-negative scalar value. For example, in our exercise, the magnitude of vector \(\mathbf{u}+3\mathbf{v} = \langle 1,4,1 \rangle\) is obtained as: \[ |\langle 1,4,1 \rangle| = \sqrt{1^2 + 4^2 + 1^2} = \sqrt{1 + 16 + 1} = \sqrt{18} \]

• Magnitude helps determine vector strength or size.
• It enables comparison of distances or sizes in practical applications.

Keep in mind, the magnitude is always non-negative. Regardless of the vector's direction or components, the length is just a measure of its size.