Problem 30 - APEX Calculus

Question

Vectors \(\vec{u}\) and \(\vec{v}\) are given. Write \(\vec{u}\) as the sum of two vectors, one of which is parallel to \(\vec{v}\) and one of which is perpendicular to \(\vec{v}\). Note: these are the same pairs of vectors as found in Exercises 21-26. \(\vec{u}=\langle-3,2\rangle, \vec{v}=\langle 2,3\rangle\)

Step-by-Step Solution

Answer

\( \vec{u} = \langle 0, 0 \rangle + \langle -3, 2 \rangle \)

Step 1: Write the formula for the projection

The vector \(\vec{u}\) can be expressed as a sum of two vectors: one parallel to \(\vec{v}\) and one perpendicular to \(\vec{v}\). The parallel component, \(\vec{u}_{\parallel}\), is the projection of \(\vec{u}\) onto \(\vec{v}\) and is given by the formula: \[\vec{u}_{\parallel} = \frac{\vec{u} \cdot \vec{v}}{\vec{v} \cdot \vec{v}} \vec{v}\]

Step 2: Compute the dot products

First, we calculate the dot product \(\vec{u} \cdot \vec{v}\): \[\vec{u} \cdot \vec{v} = (-3)(2) + (2)(3) = -6 + 6 = 0\] Now calculate \(\vec{v} \cdot \vec{v}\): \[\vec{v} \cdot \vec{v} = (2)(2) + (3)(3) = 4 + 9 = 13\]

Step 3: Calculate the parallel component

Using the results of the dot products, we find the parallel component: \[\vec{u}_{\parallel} = \frac{0}{13} \langle 2, 3 \rangle = \langle 0, 0 \rangle\]

Step 4: Find the perpendicular component

The perpendicular component, \(\vec{u}_{\perp}\), is \(\vec{u} - \vec{u}_{\parallel}\): \[\vec{u}_{\perp} = \langle -3, 2 \rangle - \langle 0, 0 \rangle = \langle -3, 2 \rangle\]

Step 5: Verify the decomposition

Now we have \(\vec{u}\) written as a sum: \[\vec{u} = \vec{u}_{\parallel} + \vec{u}_{\perp} = \langle 0, 0 \rangle + \langle -3, 2 \rangle = \langle -3, 2 \rangle\] These vectors sum to \(\vec{u}\), confirming the decomposition.

Key Concepts

Parallel Vectors

Understanding parallel vectors is key to many vector operations. Two vectors are parallel if they have the same or exactly opposite direction. This means one vector is just a scalar multiple of the other. In the provided exercise, the task was to find the vector that is parallel to \( \vec{v} \).
Since we know from the solution that the dot product \( \vec{u} \cdot \vec{v} \) is zero, the parallel component \( \vec{u}_{\parallel} \) turned out to be the zero vector \( \langle 0, 0 \rangle \), indicating no portion of \( \vec{u} \) is in the direction of \( \vec{v} \).

Key things to know about parallel vectors:

• If two vectors are parallel, their dot product won't be zero unless one of the vectors is a zero vector.
• Parallel vectors maintain a constant angle, which is either \( 0^\circ \) or \( 180^\circ \).
• A zero vector is technically parallel to any vector, as it has no direction of its own.

Recognizing parallel vectors simplifies solving vector decomposition problems.

Perpendicular Vectors

Perpendicular vectors form a 90-degree angle with each other. This unique property means their dot product is zero.
In the exercise, the vector \( \vec{u}_{\perp} \) is perpendicular to \( \vec{v} \), confirmed since the dot product \( \vec{u} \cdot \vec{v} \) equals zero.

Understanding when vectors are perpendicular helps in many areas:

• If two vectors are perpendicular, it implies they are orthogonal, having no influence on each other's direction.
• This orthogonality simplifies many vector calculations by turning interactions into zeros due to the dot product.
• Perpendicular vectors can be located by finding the vector that satisfies the zero dot product condition with the given vector.

This concept is essential for understanding projections and decomposing vectors.

Dot Product

The dot product, also known as the scalar product, combines two vectors to produce a scalar. Its formula for vectors \( \vec{a} = \langle a_1, a_2 \rangle \) and \( \vec{b} = \langle b_1, b_2 \rangle \) is:\[\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2\]
In our exercise, calculating \( \vec{u} \cdot \vec{v} \) gives 0, implying a perpendicular relationship.

Importance of the dot product includes:

• The sign of the dot product indicates the nature of the angle between vectors: if positive, less than 90 degrees; if zero, exactly 90 degrees; if negative, more than 90 degrees.
• It allows quantifying the degree of parallelism between vectors, crucial in decomposition.
• The dot product is also useful in determining lengths and angles of vectors in more complex geometric contexts.

Mastering the dot product is essential for understanding vector operations and the geometries of vector spaces.