Problem 4 - Advanced Engineering Mathematics

Question

In Problems \(1-12, \mathbf{a}=\langle 2,-3,4\rangle, \mathbf{b}=\langle-1,2,5\rangle\), and \(\mathbf{c}=\langle 3,6,-1\rangle .\) Find the indicated scalar or vector. \(\mathbf{a} \cdot \mathbf{( b}+\mathbf{c})\)

Step-by-Step Solution

Answer

The result of \( \mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) \) is \(-4\).

Step 1: Calculate \( \mathbf{b} + \mathbf{c} \)

To find \( \mathbf{b} + \mathbf{c} \), we need to add the corresponding components of the vectors \( \mathbf{b} = \langle -1, 2, 5 \rangle \) and \( \mathbf{c} = \langle 3, 6, -1 \rangle \). Calculate: \[ (-1 + 3, 2 + 6, 5 + (-1)) = \langle 2, 8, 4 \rangle \] Thus, \( \mathbf{b} + \mathbf{c} = \langle 2, 8, 4 \rangle \).

Step 2: Calculate the Dot Product \( \mathbf{a} \cdot \mathbf{(b+c)} \)

Now we need to find the dot product of \( \mathbf{a} = \langle 2, -3, 4 \rangle \) and \( \mathbf{b} + \mathbf{c} = \langle 2, 8, 4 \rangle \). The dot product formula is:\[ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \]Applying this to our vectors:\[ (2 \times 2) + (-3 \times 8) + (4 \times 4) = 4 - 24 + 16 \]Calculating the terms:\[ 4 - 24 + 16 = -4 \]Thus, \( \mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = -4 \).

Key Concepts

Dot Product

In vector algebra, the dot product is an operation that takes two vectors and returns a scalar, representing their multiplication. It provides a measure of how much one vector extends in the direction of another. To calculate the dot product of two vectors \( \mathbf{a} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{b} = \langle b_1, b_2, b_3 \rangle \), use the formula:

• \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \)

This means you multiply corresponding components of the two vectors and sum the results. In the exercise, the vectors are given by \( \mathbf{a} = \langle 2, -3, 4 \rangle \) and \( \mathbf{b+c} = \langle 2, 8, 4 \rangle \). Applying the formula:

• The x-components: \( 2 \times 2 = 4 \)
• The y-components: \( -3 \times 8 = -24 \)
• The z-components: \( 4 \times 4 = 16 \)

Finally, sum these: \( 4 - 24 + 16 = -4 \). Thus, the dot product \( \mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) \) equals -4. The negative result indicates that the two vectors are oriented in opposite directions to some degree.

Vector Addition

Vector addition is a fundamental operation where two vectors are combined to give a resultant vector. This operation involves adding each corresponding component of the vectors. For vectors \( \mathbf{b} = \langle -1, 2, 5 \rangle \) and \( \mathbf{c} = \langle 3, 6, -1 \rangle \), vector addition is done as follows:

• Add the x-components: \( -1 + 3 = 2 \)
• Add the y-components: \( 2 + 6 = 8 \)
• Add the z-components: \( 5 + (-1) = 4 \)

Hence, the sum of these vectors is \( \mathbf{b} + \mathbf{c} = \langle 2, 8, 4 \rangle \).
Vector addition is used in physical applications, like determining the resultant force or displacement when multiple vectors act together.

Scalar Multiplication

Scalar multiplication involves multiplying each component of a vector by a constant, called a scalar. This process changes the magnitude of the vector but not its direction unless the scalar is negative, which would also reverse it.

• If \( k \) is the scalar and \( \mathbf{v} = \langle x, y, z \rangle \), then \( k \mathbf{v} = \langle kx, ky, kz \rangle \).

For instance, multiplying the vector \( \mathbf{u} = \langle 2, 3, -1 \rangle \) by a scalar \( k = 3 \) would result in:

• \( 3 \mathbf{u} = \langle 6, 9, -3 \rangle \)

Understanding scalar multiplication is crucial in scaling vectors in physics, engineering, and computer graphics, where adjusting the magnitude of vectors is often required.