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. \(|\mathbf{u}+3 \mathbf{v}|\) $$\mathbf{u}=(-2,-3,0), \mathbf{v}=\langle 1,2,1\rangle$$

Short answer

**Question:** Given the vectors \(\mathbf{u} = (-2, -3, 0)\) and \(\mathbf{v} = (1, 2, 1)\), compute the following expressions: a) \(3\mathbf{u} + 2\mathbf{v}\) b) \(4\mathbf{u} - \mathbf{v}\) c) \(|\mathbf{u}+3\mathbf{v}|\) **Answer:** a) \(-4, -5, 2\) b) \(-9, -14, -1\) c) \(\sqrt{19}\)

Step-by-step solution

Part a: Find \(3\mathbf{u} + 2\mathbf{v}\)

To find \(3\mathbf{u}+2\mathbf{v}\), first multiply each vector by their respective scalars, and then add the results component-wise. $$3\mathbf{u} = 3(-2,-3,0) = (-6,-9,0)$$ $$2\mathbf{v} = 2(1,2,1) = (2,4,2)$$ Now, add the scaled vectors component-wise: $$(3\mathbf{u}+2\mathbf{v})=(-6,-9,0)+(2,4,2)=(-6+2,-9+4,0+2)=(-4,-5,2)$$

Part b: Find \(4\mathbf{u} - \mathbf{v}\)

To find \(4\mathbf{u}-\mathbf{v}\), first multiply vector \(\mathbf{u}\) by the scalar \(4\), and then subtract vector \(\mathbf{v}\) component-wise. $$4\mathbf{u} = 4(-2,-3,0) = (-8,-12,0)$$ Now, subtract \(\mathbf{v}\) component-wise: $$(4\mathbf{u}-\mathbf{v})=(-8,-12,0)-(1,2,1) =(-8-1,-12-2,0-1) =(-9,-14,-1)$$

Part c: Find \(|\mathbf{u}+3\mathbf{v}|\)

To find \(|\mathbf{u}+3\mathbf{v}|\), first multiply vector \(\mathbf{v}\) by the scalar \(3\), and then add the resulting vector to \(\mathbf{u}\). Finally, find the magnitude of the sum: $$3\mathbf{v} = 3(1,2,1) = (3,6,3)$$ $$\mathbf{u} + 3\mathbf{v} = (-2,-3,0)+(3,6,3) = (-2+3,-3+6,0+3) = (1,3,3)$$ Now, find the magnitude of \((1,3,3)\) using the Pythagorean theorem in three dimensions: $$|\mathbf{u}+3\mathbf{v}| = \sqrt{(1)^2+(3)^2+(3)^2} = \sqrt{1+9+9} = \sqrt{19}$$

Key concepts

Scalar Multiplication

Scalar multiplication is a fundamental operation in vector algebra that involves multiplying a vector by a scalar (a real number). Imagine stretching or compressing a vector without changing its direction, just its length. Here's how it works:

• To multiply a vector by a scalar, multiply each component of the vector by the scalar.
• This operation changes the magnitude (or length) of the vector but keeps its direction the same if the scalar is positive. If the scalar is negative, the direction reverses.

For example, let's look at finding \(3\mathbf{u}\) with \(\mathbf{u} = (-2, -3, 0)\):

• Multiply each component: \(3(-2, -3, 0) = (-6, -9, 0)\).
• This scales \(\mathbf{u}\) by three, stretching it without altering its direction.

Remember that scalar multiplication is as simple as multiplying each component separately. This basic operation is a building block for more complex vector manipulations.

Vector Addition

Vector addition is about combining two vectors into one, which we call the resultant vector. This operation is like following the directions of two vectors successively. Here's how to add vectors:

• Ensure both vectors have the same number of components.
• Add corresponding components from each vector to compute the resultant vector's components.

For instance, consider \(3\mathbf{u} + 2\mathbf{v}\) where \(\mathbf{u} = (-2, -3, 0)\) and \(\mathbf{v} = (1, 2, 1)\):

• First, scale the vectors: \(3\mathbf{u} = (-6, -9, 0)\) and \(2\mathbf{v} = (2, 4, 2)\).
• Next, add them component-wise: \[(-6, -9, 0) + (2, 4, 2) = (-6+2, -9+4, 0+2) = (-4, -5, 2)\].

This resultant vector \((-4, -5, 2)\) represents the total direction and magnitude of the two original movements when applied in sequence. Vector addition allows us to compile directions from different influences into a single, concise vector path.

Magnitude of a Vector

The magnitude of a vector, often viewed as its length, is a measurement of how far the vector extends in space. Calculate the magnitude using the Pythagorean theorem, extended into three-dimensional space:

• Square each component of the vector.
• Add these squares together.
• Take the square root of this sum to find the magnitude.

Let's look at the example of finding the magnitude of \(\mathbf{u}+3\mathbf{v}\) where \(\mathbf{u} = (-2, -3, 0)\) and \(3\mathbf{v} = (3, 6, 3)\) yields \( (1, 3, 3) \):

• Combine the vectors: \( (1, 3, 3) = (-2+3, -3+6, 0+3) \).
• Calculate the magnitude: \[|\mathbf{u}+3\mathbf{v}| = \sqrt{1^2 + 3^2 + 3^2} = \sqrt{1 + 9 + 9} = \sqrt{19}\].

This value, \(\sqrt{19}\), represents the overall "size" of the vector in space, disregarding its direction. Understanding vector magnitude is vital in physics and engineering, where knowing both the size and direction of physical quantities is crucial.