Problem 21 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-28: Given that \(a=\langle 2,-3\rangle, \quad b=\langle 3,4\rangle\), and \(c=\langle-1,5\rangle\), find the number. (a) \(\mathbf{a} \cdot(\mathbf{b}+\mathbf{c})\) (b) \(\mathbf{a} \cdot \mathbf{b}+\mathbf{a} \cdot \mathbf{c}\)

Step-by-Step Solution

Answer

Both (a) and (b) are -23.

Step 1: Add vectors \(\mathbf{b}\) and \(\mathbf{c}\)

First, we add vectors \(\mathbf{b}\) and \(\mathbf{c}\) by adding their corresponding components together. The resultant vector \(\mathbf{b} + \mathbf{c}\) is calculated as follows:\[\mathbf{b} + \mathbf{c} = \langle 3, 4 \rangle + \langle -1, 5 \rangle = \langle 3 + (-1), 4 + 5 \rangle = \langle 2, 9 \rangle\]

Step 2: Calculate dot product of \(\mathbf{a}\) and \(\mathbf{b}+\mathbf{c}\)

Now, compute the dot product of vector \(\mathbf{a}\) with the resultant vector from Step 1, \(\mathbf{a} \cdot (\mathbf{b} + \mathbf{c})\):\[\mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = \langle 2, -3 \rangle \cdot \langle 2, 9 \rangle\]\[= 2 \times 2 + (-3) \times 9 = 4 - 27 = -23\]

Step 3: Calculate dot product of \(\mathbf{a}\) and \(\mathbf{b}\)

Calculate the dot product of \(\mathbf{a}\) and \(\mathbf{b}\):\[\mathbf{a} \cdot \mathbf{b} = \langle 2, -3 \rangle \cdot \langle 3, 4 \rangle\]\[= 2 \times 3 + (-3) \times 4 = 6 - 12 = -6\]

Step 4: Calculate dot product of \(\mathbf{a}\) and \(\mathbf{c}\)

Calculate the dot product of \(\mathbf{a}\) and \(\mathbf{c}\):\[\mathbf{a} \cdot \mathbf{c} = \langle 2, -3 \rangle \cdot \langle -1, 5 \rangle\]\[= 2 \times (-1) + (-3) \times 5 = -2 - 15 = -17\]

Step 5: Add results of Steps 3 and 4

Finally, add the results from Steps 3 and 4 to find \(\mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c}\):\[\mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c} = -6 + (-17) = -23\]

Key Concepts

Vector Addition

Vector addition is a core concept in vector mathematics and it's a crucial operation when dealing with vectors. When we talk about the addition of vectors, we refer to the process of combining two vectors to form a new resultant vector.

The beauty of vector addition lies in its simplicity—it follows the "head-to-tail" rule where one vector’s tail is placed at the head of the other vector. However, in terms of coordinate operations, vector addition boils down to a straightforward component-wise operation.

• Take the given vectors, in this case, \( \mathbf{b} = \langle 3, 4 \rangle \) and \( \mathbf{c} = \langle -1, 5 \rangle \).
• Add the components: the x-components together, and the y-components together.
• The sum of these components forms the resultant vector: \( \mathbf{b} + \mathbf{c} = \langle 2, 9 \rangle \).

This operation is fundamental in both two-dimensional and three-dimensional spaces, providing the first step to solving more complex vector problems.

Component-wise Operation

In vectors, the term component-wise operation is widely used to explain operations that are performed separately on corresponding components of vectors.

Whether it's addition, subtraction, or even more complex operations like the dot product, breaking the process down into component-wise steps is key.

• For vector operations, treat each vector as a list of numbers or components.
• Perform operations separately on the corresponding parts. For addition, each vector component is added to its matching component from the other vector.
• This means if you have two vectors, \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \), the component-wise operation for addition will be \( a_1 + b_1 \) and \( a_2 + b_2 \).

This division of a vector into its components makes numerous vector operations straightforward since each pair of corresponding components is treated independently. Hence, the resultant operations on vectors can be easily computed and understood.

Algebraic Manipulation

Algebraic manipulation is a powerful tool when dealing with vectors, especially in multiplying vectors using the dot product. It lets you simplify expressions and find precise solutions.

The dot product, which combines two vectors algebraically into a single number, is inherently a component operation but follows a specific algebraic pattern:

• Multiply corresponding components of the vectors. For instance, given \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \), the dot product is calculated as \( a_1b_1 + a_2b_2 \).
• Sum the results of these multiplications to obtain a scalar (a single number).

In the exercise, when you see calculations like \( \mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) \), what happens first is the vector addition \( \mathbf{b} + \mathbf{c} \), and then algebraic manipulation is used to find an accurate dot product. This order of operations ensures that vector calculations align with algebraic principles, making it easier to manage and reduce errors.