Problem 72 - Precalculus with Calculus Previews

Question

In Problems \(65-76, \mathbf{u}=\langle 1,-3,2\rangle, \mathbf{v}=\langle-1,1,1\rangle\), and \(\mathbf{w}=\langle 2,6,9\rangle .\) Find the indicated vector or scalar. $$ |\mathbf{v}| \mathbf{u}+|\mathbf{u}| \mathbf{v} $$

Step-by-Step Solution

Answer

\(\langle \sqrt{3} - \sqrt{14}, -3\sqrt{3} + \sqrt{14}, 2\sqrt{3} + \sqrt{14} \rangle\)

Step 1: Calculate Magnitude of \(\mathbf{v}\)

The magnitude of a vector \(\mathbf{v} = \langle a, b, c \rangle\) is calculated using the formula \( |\mathbf{v}| = \sqrt{a^2 + b^2 + c^2} \). For \(\mathbf{v} = \langle -1, 1, 1 \rangle\): \[ |\mathbf{v}| = \sqrt{(-1)^2 + 1^2 + 1^2} = \sqrt{1 + 1 + 1} = \sqrt{3} \] Thus, \(|\mathbf{v}| = \sqrt{3}\).

Step 2: Calculate Magnitude of \(\mathbf{u}\)

The magnitude of \(\mathbf{u}\) is calculated in the same way: \( |\mathbf{u}| = \sqrt{1^2 + (-3)^2 + 2^2} = \sqrt{1 + 9 + 4} = \sqrt{14} \). So, \(|\mathbf{u}| = \sqrt{14}\).

Step 3: Scale Vector \(\mathbf{u}\) by \( |\mathbf{v}| \)

We need to multiply each component of \(\mathbf{u}\) by \(|\mathbf{v}|\): \[ |\mathbf{v}| \mathbf{u} = \sqrt{3} \langle 1, -3, 2 \rangle = \langle \sqrt{3} \cdot 1, \sqrt{3} \cdot -3, \sqrt{3} \cdot 2 \rangle = \langle \sqrt{3}, -3\sqrt{3}, 2\sqrt{3} \rangle \]

Step 4: Scale Vector \(\mathbf{v}\) by \( |\mathbf{u}| \)

Next, multiply each component of \(\mathbf{v}\) by \(|\mathbf{u}|\): \[ |\mathbf{u}| \mathbf{v} = \sqrt{14} \langle -1, 1, 1 \rangle = \langle \sqrt{14} \cdot -1, \sqrt{14} \cdot 1, \sqrt{14} \cdot 1 \rangle = \langle -\sqrt{14}, \sqrt{14}, \sqrt{14} \rangle \]

Step 5: Sum Scaled Vectors

Now, add the vectors from Steps 3 and 4: \[ |\mathbf{v}|\mathbf{u} + |\mathbf{u}|\mathbf{v} = \langle \sqrt{3}, -3\sqrt{3}, 2\sqrt{3} \rangle + \langle -\sqrt{14}, \sqrt{14}, \sqrt{14} \rangle \] Perform the addition component-wise: \[ \langle \sqrt{3} - \sqrt{14}, -3\sqrt{3} + \sqrt{14}, 2\sqrt{3} + \sqrt{14} \rangle \]

Step 6: Complete the Solution

Therefore, the resulting vector is: \(\langle \sqrt{3} - \sqrt{14}, -3\sqrt{3} + \sqrt{14}, 2\sqrt{3} + \sqrt{14} \rangle\).

Key Concepts

Vector Magnitude

Understanding vector magnitude is essential to grasp vector operations. Picture vectors as arrows in space, where magnitude represents the arrow's length. Calculating magnitude helps quantify the size of a vector. The formula for finding the magnitude of a vector \(\mathbf{v} = \langle a, b, c \rangle\) is: \[ \|\mathbf{v}\| = \sqrt{a^2 + b^2 + c^2} \] This equation uses the components of the vector, squaring each before summing and taking the square root. It's akin to using the Pythagorean theorem in three dimensions. For example, to find the magnitude of \(\mathbf{v} = \langle -1, 1, 1 \rangle\), calculate:

• Square each component: \((-1)^2 = 1, 1^2 = 1, 1^2 = 1\)
• Add them up: \(1 + 1 + 1 = 3\)
• Take the square root of the sum: \(\sqrt{3}\)

Thus, the magnitude \(\|\mathbf{v}\|\) is \(\sqrt{3}\). Magnitude plays a crucial role in operations like scaling vectors, as it reflects how large the vector is in space.

Scalar Multiplication

Scalar multiplication involves resizing a vector by multiplying it with a scalar, a real number. Imagine increasing or decreasing the vector's length without altering its direction. Every vector component is multiplied by the scalar, hence affecting the vector's magnitude. Let's explore how to perform scalar multiplication using the example: scaling \(\mathbf{u} = \langle 1, -3, 2 \rangle\) by \(\|\mathbf{v}\| = \sqrt{3}\). The operation involves:

• Multiplying the first component: \(\sqrt{3} \times 1 = \sqrt{3}\)
• Multiplying the second component: \(\sqrt{3} \times -3 = -3\sqrt{3}\)
• Multiplying the third component: \(\sqrt{3} \times 2 = 2\sqrt{3}\)

The results provide a new vector: \(\langle \sqrt{3}, -3\sqrt{3}, 2\sqrt{3} \rangle\). It maintains the direction of \(\mathbf{u}\) but adjusts its length to \(\sqrt{3}\) times its original size. Scalar multiplication is also foundational for scaling vectors in various mathematical computations.

Vector Addition

Vector addition is when you combine two vectors to find their resultant vector. This operation often gives you a new vector reflecting both direction and combined magnitude effects. When adding vectors, you sum their corresponding components. Consider adding two vectors such as \(|\mathbf{v}|\mathbf{u} = \langle \sqrt{3}, -3\sqrt{3}, 2\sqrt{3} \rangle\) and \(|\mathbf{u}|\mathbf{v} = \langle -\sqrt{14}, \sqrt{14}, \sqrt{14} \rangle\):

• The first component: \(\sqrt{3} + (-\sqrt{14}) = \sqrt{3} - \sqrt{14}\)
• The second component: \(-3\sqrt{3} + \sqrt{14}\)
• The third component: \(2\sqrt{3} + \sqrt{14}\)

Each component is added separately, creating the resultant vector \(\langle \sqrt{3} - \sqrt{14}, -3\sqrt{3} + \sqrt{14}, 2\sqrt{3} + \sqrt{14} \rangle\). This vector represents both input vectors acting together. Vector addition is crucial for navigating different spaces, calculating forces, and solving physics problems.