Problem 22 - APEX Calculus

Question

Vectors \(\vec{u}\) and \(\vec{v}\) are given. Find \(\operatorname{proj}_{\vec{v}} \vec{u}\) the orthogonal projection of \(\vec{u}\) onto \(\vec{v},\) and sketch all three vectors with the same initial point. \(\vec{u}=\langle 5,5\rangle, \vec{v}=\langle 1,3\rangle\)

Step-by-Step Solution

Answer

The orthogonal projection of \( \vec{u} \) onto \( \vec{v} \) is \( \langle 2, 6 \rangle \).

Step 1: Understand the Formula

The orthogonal projection of vector \( \vec{u} \) onto vector \( \vec{v} \), denoted as \( \operatorname{proj}_{\vec{v}} \vec{u} \), is calculated by the formula: \[ \operatorname{proj}_{\vec{v}} \vec{u} = \frac{\vec{u} \cdot \vec{v}}{\vec{v} \cdot \vec{v}} \vec{v} \]where \( \vec{u} \cdot \vec{v} \) is the dot product of the two vectors, and \( \vec{v} \cdot \vec{v} \) is the dot product of \( \vec{v} \) with itself.

Step 2: Compute the Dot Product \( \vec{u} \cdot \vec{v} \)

Calculate the dot product of \( \vec{u} \) and \( \vec{v} \):\[ \vec{u} \cdot \vec{v} = (5)(1) + (5)(3) = 5 + 15 = 20 \]

Step 3: Compute the Dot Product \( \vec{v} \cdot \vec{v} \)

Calculate the dot product of \( \vec{v} \) with itself:\[ \vec{v} \cdot \vec{v} = (1)(1) + (3)(3) = 1 + 9 = 10 \]

Step 4: Substitute in the Projection Formula

Substitute the results from Steps 2 and 3 into the projection formula:\[ \operatorname{proj}_{\vec{v}} \vec{u} = \frac{20}{10} \vec{v} = 2 \vec{v} \]Thus, \( \operatorname{proj}_{\vec{v}} \vec{u} = 2 \langle 1, 3 \rangle = \langle 2, 6 \rangle \).

Step 5: Sketch the Vectors

Draw vectors \( \vec{u} = \langle 5,5 \rangle \), \( \vec{v} = \langle 1,3 \rangle \), and \( \operatorname{proj}_{\vec{v}} \vec{u} = \langle 2,6 \rangle \) starting from the same initial point. Use a coordinate plane to help visualize their direction and relative size.

Key Concepts

Dot Product

The dot product is a fundamental concept used in vector mathematics. It allows us to multiply two vectors to get a scalar. It's particularly helpful when working in a plane to determine the angle between vectors or projection operations. To compute the dot product of two vectors, you multiply their corresponding components and then add up those products. For example, if you have vectors \(\vec{u} = \langle a, b \rangle\) and \(\vec{v} = \langle c, d \rangle\), the dot product is calculated as:

• \(\vec{u} \cdot \vec{v} = a \cdot c + b \cdot d\)

In our specific example, the dot product of \(\vec{u} = \langle 5,5 \rangle\) and \(\vec{v} = \langle 1,3 \rangle\) was found by calculating \((5 \cdot 1) + (5 \cdot 3) = 20\). Calculating the dot product is a critical step for determining vector projections.

Orthogonal Projection

Orthogonal projection is about casting one vector onto another referring vector, much like how a light casts a shadow of a figure onto the ground. This operation helps us identify what part of one vector lies in the direction of another. The formula for orthogonal projection is given by:

• \(\operatorname{proj}_{\vec{v}} \vec{u} = \frac{\vec{u} \cdot \vec{v}}{\vec{v} \cdot \vec{v}} \vec{v}\)

In our exercise, we calculated the orthogonal projection of \(\vec{u}\) onto \(\vec{v}\) using the previously computed dot products. First was \(20\) from the product of \(\vec{u}\) and \(\vec{v}\), and then the product of \(\vec{v}\) with itself, which was 10. When we substitute these values into our formula, we get:

• \(\operatorname{proj}_{\vec{v}} \vec{u} = \frac{20}{10} \vec{v} = 2 \vec{v}\)

Thus, the resulting orthogonal projection vector is \(\langle 2, 6 \rangle\). It represents the component of \(\vec{u}\) that points in the same direction as \(\vec{v}\) and is useful in resolving vector components.

Vectors in the Plane

Vectors in a plane are represented by coordinates that show direction and magnitude within a two-dimensional space. For example, a vector \(\vec{u} = \langle x, y \rangle\) describes a movement from the origin \((0,0)\) to \((x,y)\). These vectors are used to model real-world quantities like force and velocity. When sketching vectors such as \(\vec{u} = \langle 5,5 \rangle\) and \(\vec{v} = \langle 1,3 \rangle\), their direction is shown by arrows starting at the same initial point. You can draw these on a coordinate plane, using arrows to indicate the vector directions and lengths.

• Length of vector \(\vec{u}\) is \(\sqrt{5^2 + 5^2} = \sqrt{50}\)
• Direction of each vector can be interpreted in relation to the axes

Here, you would plot the vectors from the origin and draw \(\operatorname{proj}_{\vec{v}} \vec{u} = \langle 2,6 \rangle\), demonstrating the projection visually. This process solidifies understanding of how vectors interact and resolve into components on a plane.