Problem 95 - Algebra and Trigonometry

Question

Let \(\mathbf{v}=2 \mathbf{i}-5 \mathbf{j}\) and \(\mathbf{w}=-4 \mathbf{i}+3 \mathbf{j} .\) Find each of the following. a) \(\mathbf{v}+\mathbf{w}\) b) \(2 \mathbf{v}-\mathbf{w}\) c) \(\mathbf{v} \cdot \mathbf{w}\) d) \(\mathbf{v} \cdot \mathbf{v}\)

Step-by-Step Solution

Answer

a) \(\mathbf{v} + \mathbf{w} = -2\mathbf{i} - 2\mathbf{j}\) b) \(2\mathbf{v} - \mathbf{w} = 8\mathbf{i} - 13\mathbf{j}\) c) \(\mathbf{v} \cdot \mathbf{w} = -23\) d) \(\mathbf{v} \cdot \mathbf{v} = 29\)

Step 1: Find Vector Addition \(\mathbf{v} + \mathbf{w}\)

The vectors \(\mathbf{v}\) and \(\mathbf{w}\) are given as \(2\mathbf{i} - 5\mathbf{j}\) and \(-4\mathbf{i} + 3\mathbf{j}\) respectively. We add the corresponding \(\mathbf{i}\) and \(\mathbf{j}\) components from each vector together. So, \(\mathbf{v}+\mathbf{w}\) = \((2 - 4)\mathbf{i} + (-5 + 3)\mathbf{j}\) = \(-2\mathbf{i} - 2\mathbf{j}\)

Step 2: Find Scalar Multiplication and Subtraction \(2\mathbf{v} - \mathbf{w}\)

Now, we compute the scalar multiplication of \(\mathbf{v}\) by two and then subtract \(\mathbf{w}\) from the result. So, \(2\mathbf{v} - \mathbf{w}\) = \(2 * (2\mathbf{i} - 5\mathbf{j}) - (-4\mathbf{i} + 3\mathbf{j})\) = \((4\mathbf{i} - 10\mathbf{j}) - (-4\mathbf{i} + 3\mathbf{j})\) = \(4\mathbf{i} + 4\mathbf{i} - 10\mathbf{j} - 3\mathbf{j}\) = \(8\mathbf{i} - 13\mathbf{j}\)

Step 3: Compute Dot Product \(\mathbf{v} \cdot \mathbf{w}\)

The dot product of two vectors is obtained by multiplying their corresponding components and then adding the results. So, \(\mathbf{v} \cdot \mathbf{w}\) = \((2)(-4) + (-5)(3)\) = \(-8 - 15\) = \(-23\)

Step 4: Calculate Dot Product \(\mathbf{v} \cdot \mathbf{v}\)

It is similar to step 3, but now we find the dot product of \(\mathbf{v}\) with itself. So, \(\mathbf{v} \cdot \mathbf{v}\) = \((2)(2) + (-5)(-5)\) = \(4 + 25\) = \(29\).

Key Concepts

Vector Addition

Vector addition involves combining two vectors to form a new vector. If you have two vectors, each with components represented by letters and numbers, you simply add each corresponding component together. In our example, the vectors \(\mathbf{v} = 2\mathbf{i} - 5\mathbf{j}\) and \(\mathbf{w} = -4\mathbf{i} + 3\mathbf{j}\) are added component-wise.

• The \(\mathbf{i}\) components are \(2\) and \(-4\), which add to \(2 + (-4) = -2\).
• The \(\mathbf{j}\) components are \(-5\) and \(3\), adding to \(-5 + 3 = -2\).

Thus, the sum of the vectors is \(-2\mathbf{i} - 2\mathbf{j}\). This new vector points to a different location in space. Remember, the direction and length of the new vector depend on these summed components.

Scalar Multiplication

Scalar multiplication means multiplying a vector by a scalar (a real number). This operation changes the magnitude of the vector but not its direction. For instance, multiplying vector \(\mathbf{v} = 2\mathbf{i} - 5\mathbf{j}\) by the scalar \(2\) results in a new vector:

• Multiply the \(\mathbf{i}\) component: \(2 \times 2 = 4\).
• Multiply the \(\mathbf{j}\) component: \(2 \times (-5) = -10\).

The resulting vector \(2\mathbf{v}\) is \(4\mathbf{i} - 10\mathbf{j}\). It's like scaling the vector by \(2\), making it twice as long while pointing in the same direction.

Dot Product

The dot product is an operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. It reflects how much one vector goes in the direction of another. For vectors \(\mathbf{v} = 2\mathbf{i} - 5\mathbf{j}\) and \(\mathbf{w} = -4\mathbf{i} + 3\mathbf{j}\), the dot product is calculated by multiplying corresponding components and summing them up:

• Multiply \(2\) and \(-4\): \(2 \times -4 = -8\).
• Multiply \(-5\) and \(3\): \(-5 \times 3 = -15\).

Adding these results gives us \(-8 + (-15) = -23\). This negative result suggests that the vectors point in somewhat opposite directions.

Vector Subtraction

Subtracting one vector from another is similar to vector addition, but here you subtract each component of the second vector from the first. For the calculation of \(2\mathbf{v} - \mathbf{w}\), start by finding \(2\mathbf{v}\) (already calculated in scalar multiplication as \(4\mathbf{i} - 10\mathbf{j}\)) and then subtracting \(\mathbf{w}\):

• \(\mathbf{i}\) components: \(4 - (-4) = 8\).
• \(\mathbf{j}\) components: \(-10 - 3 = -13\).

The result is \(8\mathbf{i} - 13\mathbf{j}\). Vector subtraction finds the difference between vectors, resulting in a new vector that points from the tip of one to the tip of the other, effectively illustrating a shift in position.