Problem 27 - Algebra and Trigonometry

Question

In Exercises 21–38, let. $$\mathbf{u}=2 \mathbf{i}-5 \mathbf{j}, \mathbf{v}=-3 \mathbf{i}+7 \mathbf{j}, \text { and } \mathbf{w}=-\mathbf{i}-6 \mathbf{j}$$ Find each specified vector or scalar. $$5 \mathbf{v}$$

Step-by-Step Solution

Answer

The result of multiplying the vector \( \mathbf{v} \) by the scalar 5 is \( -15 \mathbf{i} + 35 \mathbf{j} \).

Step 1: Identify the Components of the Vector

Vector \( \mathbf{v} \) has components -3 and 7, corresponding to \( \mathbf{i} \) and \( \mathbf{j} \) respectively. Therefore, vector \( \mathbf{v} \) can be written as \( \mathbf{v} = -3 \mathbf{i} + 7 \mathbf{j} \).

Step 2: Multiply the Vector by the Scalar

Next, multiply each component of \( \mathbf{v} \) by the scalar quantity 5. This gives \( 5 \mathbf{v} = 5(-3 \mathbf{i} + 7 \mathbf{j}) = -15 \mathbf{i} + 35 \mathbf{j} \).

Key Concepts

Vector Components

Understanding the components of a vector is crucial when performing various operations in vector algebra. A vector in two-dimensional space is represented as \(\mathbf{u}=a\mathbf{i}+b\mathbf{j}\), where \(a\) and \(b\) are the vector's components along the horizontal \(x\)-axis and vertical \(y\)-axis, respectively. These components are essentially projections of the vector onto the axes.

In the exercise provided, the vector \(\mathbf{v}\) was represented by its components \(\mathbf{v}=-3\mathbf{i}+7\mathbf{j}\). These components \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors that point in the positive directions of the \(x\)-axis and \(y\)-axis, respectively. The coefficient of \(\mathbf{i}\) is the magnitude of \(\mathbf{v}\) in the \(x\)-direction, while the coefficient of \(\mathbf{j}\) is the magnitude of \(\mathbf{v}\) in the \(y\)-direction. Understanding this allows us to perform algebraic operations such as addition, subtraction, and in the exercise's case, scalar multiplication.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a real number (also known as a scalar). This operation scales the vector by increasing or decreasing its magnitude without altering its direction, unless the scalar is negative, in which case the direction is reversed.

From the exercise, we see that when multiplying the vector \(\mathbf{v}\) by the scalar \(5\), each component of \(\mathbf{v}\) is multiplied by \(5\): \(5\mathbf{v} = 5(-3 \mathbf{i} + 7 \mathbf{j}) = -15 \mathbf{i} + 35 \mathbf{j}\). The resultant vector has the same direction as the original \(\mathbf{v}\), but its length is \(5\) times longer. It's important to note that scalar multiplication is distributive over vector addition. This means that for any vectors \(\mathbf{a}\) and \(\mathbf{b}\) and scalars \(c\) and \(d\), the following is true: \(c(\mathbf{a} + \mathbf{b}) = c\mathbf{a} + c\mathbf{b}\). This property is used extensively in simplifying vector expressions.

Vector Algebra

Vector algebra encompasses operations such as addition, subtraction, and multiplication of vectors. When we add or subtract vectors, we're combining or separating them by aligning their tails or heads and then constructing a vector from the start of the first to the end of the second vector.

For multiplication, as seen in the exercise, we may perform scalar multiplication or find the dot product and cross product of vectors (which are not demonstrated in this problem). Vector algebra follows many of the same rules as conventional algebra, including commutative, associative, and distributive laws. However, it's important to always consider direction in addition to magnitude. For example, the sum of two vectors \(\mathbf{a}\) and \(\mathbf{b}\) can be found by algebraically adding their corresponding components: \(\mathbf{a} + \mathbf{b} = (a_1\mathbf{i}+a_2\mathbf{j}) + (b_1\mathbf{i}+b_2\mathbf{j}) = (a_1+b_1)\mathbf{i} + (a_2+b_2)\mathbf{j}\).

By mastering vector algebra, we are able to solve a diverse range of problems in physics, engineering, and advanced mathematics—especially when it comes to dealing with quantities that have both magnitude and direction.