Problem 40 - Algebra and Trigonometry

Question

Let $$\mathbf{u}=-\mathbf{i}+\mathbf{j}, \quad \mathbf{v}=3 \mathbf{i}-2 \mathbf{j}, \quad \text { and } \quad \mathbf{w}=-5 \mathbf{j}$$ Find each specified scalar or vector. $$ 4 u \cdot(5 v-3 w) $$

Step-by-Step Solution

Answer

The scalar is -40.

Step 1: Scalar Multiplication of Vectors

\(5\mathbf{v} = 5(3\mathbf{i} - 2\mathbf{j}) = 15\mathbf{i} - 10\mathbf{j}\). Scalar multiplication is performed by multiplying the scalar (5 in this case) with each component of the vector \(\mathbf{v}\).

Step 2: Scalar Multiplication of Vectors

\(3\mathbf{w} = 3(-5\mathbf{j}) = -15\mathbf{j}\). Here, scalar multiplication is performed by multiplying the scalar (3 in this case) with each component of the vector \(\mathbf{w}\).

Step 3: Vector Subtraction

\(5\mathbf{v} - 3\mathbf{w} = (15\mathbf{i} - 10\mathbf{j}) - (-15\mathbf{j}) = 15\mathbf{i} - 10\mathbf{j} + 15\mathbf{j} = 15\mathbf{i}+5\mathbf{j}\)

Step 4: Dot Product of Vectors

Calculate the dot product of \(\mathbf{u}\) and \(5\mathbf{v} - 3\mathbf{w}\). The dot product of two vectors \(\mathbf{a} = a1\mathbf{i} + a2\mathbf{j}\) and \(\mathbf{b} = b1\mathbf{i} + b2\mathbf{j}\) is given by \(a1*b1 + a2*b2\). Thus, \(4\mathbf{u}\cdot(5\mathbf{v} - 3\mathbf{w}) = 4[(-1)(15) + (1)(5)] = -40\)

Key Concepts

Scalar Multiplication

Scalar multiplication is a simple yet powerful tool in vector algebra. It involves multiplying a vector by a scalar, which is just a fancy word for an ordinary number. When you multiply a vector by a scalar, you change the vector's magnitude but not its direction. Think of it as resizing the vector.
For instance, if we take the vector \( \mathbf{v} = 3\mathbf{i} - 2\mathbf{j} \) and multiply it by the scalar 5, the result is a new vector \( 5\mathbf{v} = 15\mathbf{i} - 10\mathbf{j} \). Here, each component of the vector \( \mathbf{v} \) is multiplied by 5.
The same principle applies when multiplying \( \mathbf{w} = -5\mathbf{j} \) by the scalar 3, giving us \( 3\mathbf{w} = -15\mathbf{j} \). The vector has been scaled, making a longer version of itself, but it still points in the same direction, along the \( \mathbf{j} \) axis or line.

Vector Subtraction

Vector subtraction allows us to find the difference between two vectors, which can be thought of as finding the vector that "bridges" the gap between them. When you subtract one vector from another, you subtract each corresponding component.
For example, suppose we want to find the vector \( 5\mathbf{v} - 3\mathbf{w} \). To do this, we first perform scalar multiplications, which results in \( 5\mathbf{v} = 15\mathbf{i} - 10\mathbf{j} \) and \( 3\mathbf{w} = -15\mathbf{j} \). Next, vector subtraction is executed by subtracting \( 15\mathbf{i} - 10\mathbf{j} \) from \( -15\mathbf{j} \), which simplifies to \( 15\mathbf{i} + 5\mathbf{j} \).
This operation reflects the way we balance the components of each vector to find the resulting vector. It's like taking two arrows on a map and figuring out where one would point if it started from the tip of the other.

Dot Product

The dot product is an operation that takes two vectors and returns a scalar (a single number). It reflects how much one vector projects onto another. To calculate the dot product, multiply the corresponding components of two vectors and then add those products together.
In our exercise, we were tasked with finding \( 4\mathbf{u} \cdot (5\mathbf{v} - 3\mathbf{w}) \). First, we discover \( 5\mathbf{v} - 3\mathbf{w} = 15\mathbf{i} + 5\mathbf{j} \). Then, the dot product \( \mathbf{u} \cdot (5\mathbf{v} - 3\mathbf{w}) \) uses the formula for dot products: given vectors \( \mathbf{a} = a1\mathbf{i} + a2\mathbf{j} \) and \( \mathbf{b} = b1\mathbf{i} + b2\mathbf{j} \), \( \mathbf{a} \cdot \mathbf{b} = a1 \times b1 + a2 \times b2 \).
Thus, substituting values gives us \[ \mathbf{u} \cdot (5\mathbf{v} - 3\mathbf{w}) = (-1)\times15 + 1\times5 = -15 + 5 = -10 \], and after considering the additional scalar 4, we have \( 4 \times (-10) = -40 \).
This concept helps us understand geometric and physical scenarios, such as determining the amount of force along a direction or analyzing angles between vectors.