Problem 41 - 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. $$ \operatorname{proj}_{\mathfrak{u}}(\mathbf{v}+\mathbf{w}) $$

Step-by-Step Solution

Answer

The vector projection of \(\mathbf{v} + \mathbf{w}\) on \(\mathbf{u}\) is \(-2\mathbf{i} + 2\mathbf{j}\).

Step 1: Calculate Sum of Vectors v and w

Firstly calculate the sum of vectors \(\mathbf{v}\) and \(\mathbf{w}\). This sum is given by \(\mathbf{v} + \mathbf{w} = (3\mathbf{i}-2\mathbf{j}) + (-5\mathbf{j}) = 3\mathbf{i} - 7\mathbf{j}\).

Step 2: Apply Projection Formula

Next, apply the projection formula which is given by \(\operatorname{proj}_{\mathfrak{u}}(\mathbf{v}+\mathbf{w}) = \frac{(\mathbf{v}+\mathbf{w}) \cdot \mathbf{u}}{\mathbf{u} \cdot \mathbf{u}} \mathbf{u}\).

Step 3: Calculate Dot Product

Substitute the vectors into the projection formula and calculate the dot products. \(\operatorname{proj}_{\mathfrak{u}}(\mathbf{v}+\mathbf{w}) = \frac{(3\mathbf{i} - 7\mathbf{j}) \cdot (-\mathbf{i} + \mathbf{j})}{(-\mathbf{i} + \mathbf{j}) \cdot (-\mathbf{i} + \mathbf{j})} (-\mathbf{i} + \mathbf{j}) = \frac{-3 + 7}{1 + 1} (-\mathbf{i} + \mathbf{j})\).

Step 4: Calculate Scalar Multiplication

Simplify the expression by calculating scalar multiplication. \(\operatorname{proj}_{\mathfrak{u}}(\mathbf{v}+\mathbf{w}) = 2(-\mathbf{i} + \mathbf{j}) = -2\mathbf{i} + 2\mathbf{j}\).

Key Concepts

Dot Product

Understanding the dot product, also known as the scalar product, is essential when working with vectors. It's an operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. The dot product of two vectors \( \mathbf{a} \) and \( \mathbf{b} \) can be calculated as \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + \dots + a_nb_n \), where \(a_i\) and \(b_i\) are components of vectors \( \mathbf{a} \) and \( \mathbf{b} \) respectively.

In the context of projection, the dot product helps to determine how much one vector extends in the direction of another vector. In simple terms, it measures the 'overlap' between vectors when one is projected onto the other. For instance, if we wish to project vector \( \mathbf{v} \) onto vector \( \mathbf{u} \), calculating the dot product of \( \mathbf{v} \) and \( \mathbf{u} \) is a crucial step. The outcome is a scalar that represents the length of the projection of \( \mathbf{v} \) on \( \mathbf{u} \) when multiplied by \( \mathbf{u} \)'s unit vector.

It's important to recognize that the dot product can also inform us about the angle between the two vectors. A positive product indicates that the vectors are pointing in a generally similar direction, while a negative product shows that they are pointing in opposite directions. If the dot product is zero, the vectors are orthogonal, meaning they are at a right angle to each other.

Scalar Multiplication

Scalar multiplication involves the multiplication of a vector by a scalar (a single number), in which each component of the vector is multiplied by the scalar. This operation is important in modifying the magnitude and direction of a vector without altering its orientation. Mathematically, if we multiply a vector \( \mathbf{v} = v_1\mathbf{i} + v_2\mathbf{j} + v_3\mathbf{k} \) by a scalar \( \alpha \), the result is \( \alpha\mathbf{v} = \alpha v_1\mathbf{i} + \alpha v_2\mathbf{j} + \alpha v_3\mathbf{k} \).

In the process of finding the vector projection, once we have computed the necessary dot products, we use scalar multiplication to scale our vector \( \mathbf{u} \) by the scalar resulting from the dot products. This scaling gives us the actual projection vector \( \operatorname{proj}_\mathfrak{u}(\mathbf{v}+\mathbf{w}) \). The scalar in question acts as a coefficient, which shrinks or enlarges the vector \( \mathbf{u} \) and lines it up along its own direction to match the required projection length. This step is crucial since it successfully converts our scalar measure of length into a vector that represents the direction and magnitude of the projection.

Sum of Vectors

The sum of vectors is another fundamental operation in vector algebra. To add two vectors together, we simply add their corresponding components. If we have vectors \( \mathbf{p} = p_1\mathbf{i} + p_2\mathbf{j} \) and \( \mathbf{q} = q_1\mathbf{i} + q_2\mathbf{j} \), their sum is \( \mathbf{p} + \mathbf{q} = (p_1 + q_1)\mathbf{i} + (p_2 + q_2)\mathbf{j} \).

Vector addition is commutative, meaning that \( \mathbf{p} + \mathbf{q} = \mathbf{q} + \mathbf{p} \), and is also associative, so \( (\mathbf{p} + \mathbf{q}) + \mathbf{r} = \mathbf{p} + (\mathbf{q} + \mathbf{r}) \). When calculating the projection of one vector onto another, it is often necessary to first find the sum of two vectors before applying the projection formula. This sum represents the combined direction and magnitude of the vectors before they are projected onto another vector. It's important to perform this operation carefully, as incorrect vector addition can lead to inaccuracies in subsequent steps, such as projection calculations. In the given exercise, it is critical to sum the vectors \( \mathbf{v} \) and \( \mathbf{w} \) accurately to proceed with the correct projection of their resultant onto vector \( \mathbf{u} \) and eventually unlock the solution to the problem.