Problem 18 - Algebra and Trigonometry

Question

Find the vectors \(\mathbf{u}+\mathbf{v}, \mathbf{u}-\mathbf{v},\) and \(3 \mathbf{u}-\frac{1}{2} \mathbf{v}\) $$ \mathbf{u}=\langle a, 2 b, 3 c\rangle, \mathbf{v}=\langle- 4 a, b,-2 c\rangle $$

Step-by-Step Solution

Answer

\( \mathbf{u} + \mathbf{v} = \langle -3a, 3b, c \rangle \), \( \mathbf{u} - \mathbf{v} = \langle 5a, b, 5c \rangle \), and \( 3\mathbf{u} - \frac{1}{2}\mathbf{v} = \langle 5a, \frac{11}{2}b, 10c \rangle \).

Step 1: Add the Vectors

To find \( \mathbf{u} + \mathbf{v} \), add the corresponding components of \( \mathbf{u} \) and \( \mathbf{v} \). \( \mathbf{u} = \langle a, 2b, 3c \rangle \) and \( \mathbf{v} = \langle -4a, b, -2c \rangle \). Thus, \( \mathbf{u} + \mathbf{v} = \langle a + (-4a), 2b + b, 3c + (-2c) \rangle = \langle -3a, 3b, c \rangle \).

Step 2: Subtract the Vectors

To find \( \mathbf{u} - \mathbf{v} \), subtract the components of \( \mathbf{v} \) from \( \mathbf{u} \). Hence, \( \mathbf{u} - \mathbf{v} = \langle a - (-4a), 2b - b, 3c - (-2c) \rangle = \langle 5a, b, 5c \rangle \).

Step 3: Calculate the Scalar Multiplication and Subtract

To find \( 3\mathbf{u} - \frac{1}{2}\mathbf{v} \), first multiply each component of \( \mathbf{u} \) by 3: \( 3\mathbf{u} = \langle 3a, 6b, 9c \rangle \). Then multiply \( \mathbf{v} \) by \( \frac{1}{2} \): \( \frac{1}{2}\mathbf{v} = \langle -2a, \frac{1}{2}b, -c \rangle \). Now subtract these results: \( 3\mathbf{u} - \frac{1}{2}\mathbf{v} = \langle 3a - (-2a), 6b - \frac{1}{2}b, 9c - (-c) \rangle = \langle 5a, \frac{11}{2}b, 10c \rangle \).

Key Concepts

Vector Addition

Vector addition is a fundamental operation that involves combining two or more vectors to produce a resultant vector. To add vectors, simply add their corresponding components together. For example, given vectors \( \mathbf{u} = \langle a, 2b, 3c \rangle \) and \( \mathbf{v} = \langle -4a, b, -2c \rangle \), the process of addition will be: - Add the first components: \( a + (-4a) = -3a \) - Add the second components: \( 2b + b = 3b \) - Add the third components: \( 3c + (-2c) = c \) Thus, the resultant vector of \( \mathbf{u} + \mathbf{v} \) is \( \langle -3a, 3b, c \rangle \). This method shows that vector addition is both straightforward and systematic. Remember, vectors are added component-wise. This means you line up their respective dimensions and simply perform addition in each dimension, producing a result in the form of a new vector. Keep in mind, vector addition is commutative, meaning that \( \mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u} \). This makes vector calculations flexible and easier to manage.

Vector Subtraction

Vector subtraction works similarly to vector addition, but instead, you subtract the corresponding components of the vectors. When performing vector subtraction for two vectors \( \mathbf{u} = \langle a, 2b, 3c \rangle \) and \( \mathbf{v} = \langle -4a, b, -2c \rangle \), you proceed as follows: - Subtract the first components: \( a - (-4a) = 5a \) - Subtract the second components: \( 2b - b = b \) - Subtract the third components: \( 3c - (-2c) = 5c \) Resulting in the vector \( \mathbf{u} - \mathbf{v} = \langle 5a, b, 5c \rangle \). In vector subtraction, make sure to carefully handle signs, especially when subtracting a negative number, as it effectively turns into addition. While subtraction is not commutative, meaning \( \mathbf{u} - \mathbf{v} eq \mathbf{v} - \mathbf{u} \), it is crucial to follow the order correctly. Subtraction of vectors can be viewed as adding a vector with the negation of another, i.e., \( \mathbf{u} - \mathbf{v} = \mathbf{u} + (-\mathbf{v}) \). By treating subtraction as an extension of addition, it becomes easier to understand and apply in various contexts.

Scalar Multiplication

Scalar multiplication involves multiplying each component of a vector by a scalar (a real number). This operation changes the magnitude of the vector but not its direction unless the scalar is negative, which inverts its direction. For a vector \( \mathbf{u} = \langle a, 2b, 3c \rangle \) and a scalar \( 3 \), the scalar multiplication is: - First component: \( 3 \cdot a = 3a \) - Second component: \( 3 \cdot 2b = 6b \) - Third component: \( 3 \cdot 3c = 9c \) This results in \( 3\mathbf{u} = \langle 3a, 6b, 9c \rangle \). For another vector \( \mathbf{v} = \langle -4a, b, -2c \rangle \) and a scalar \( \frac{1}{2} \), the multiplication results in: - First component: \( \frac{1}{2} \cdot -4a = -2a \) - Second component: \( \frac{1}{2} \cdot b = \frac{1}{2}b \) - Third component: \( \frac{1}{2} \cdot -2c = -c \) Thus, \( \frac{1}{2} \mathbf{v} = \langle -2a, \frac{1}{2}b, -c \rangle \). When combining scalar multiplication with addition or subtraction, such as \( 3\mathbf{u} - \frac{1}{2}\mathbf{v} \), you first compute the scalar multiplication for each vector. Then, you perform the vector subtraction using these new vectors. Be meticulous with calculations to ensure precision, as these operations lay the foundation for more complex vector analysis.