GRAPHS

• Graph 1 — The unit circle, a fixed vector, and a vector whose direction and magnitude can be changed.

EXAMPLE — Equivalent Vectors & Standard Position

Background & Notes

Two vectors are equivalent if:

• They have the same magnitude & direction. (geometric)
• They have the same component form. (algebraic)

A vector is in standard position if its initial point is the origin \(O\).

Assumptions

Let \(P\) and \(Q\) be the points:

\[P(-3,0) \qquad Q(1,2)\]

And let \(\vec{PQ}\) be the vector from the initial point \(P\) to terminal point \(Q\).

Instructions

1. Graph

Get out your notebook, and sketch:

• The \(x\) and \(y\) coordinate axes.
• The points \(P(-3,0)\) and \(Q(1,2)\).
• The vector \(\vec{PQ}\).

2. Compute & Graph

Use the coordinates of the points \(P(-3,0)\) and \(Q(1,2)\) to compute the component form of \(\vec{PQ}\).

Use the component form of \(\vec{PQ}\) to sketch the vector \(\vec{v}\) that is:

• Equivalent to \(\vec{PQ}\).
• In standard position.

3. Reverse Direction & Compare

On your graph, sketch the vector \(\vec{QP}\).

Then, use the coordinates of the points \(P\) and \(Q\) to compute the component form of \(\vec{QP}\).

Compare \(\vec{PQ}\) and \(\vec{QP}\) geometrically (magnitudes & directions), and algebraically (components):

• Geometric Comparison — What are the relationships between the magnitudes and directions of \(\vec{PQ}\) and \(\vec{QP}\)?
• Algebraic Comparison — What is the relationship between the components of \(\vec{PQ}\) and \(\vec{QP}\)?

EXERCISE 1 — Magnitude & Direction (Unit Vectors)

Background & Notes

A unit vector \(\hat{v}\) is a vector whose magnitude equals 1:

\[| \hat{v} | = 1\]

Every unit vector \(\hat{v}\) in the plane ( \(\mathbb{R}^2\) ) has component form:

\[\hat{v} = \big< \cos \theta, \sin \theta \big>\]

If \(\vec{v} \neq \vec{0}\) is a non-zero vector, its direction vector is the unit vector \(\hat{v}\) that has the same direction as \(\vec{v}\).

Assumptions

Let \(\vec{v}\) be the vector:

\[\vec{v} = \big< 1, 1 \big>\]

In standard position.

Instructions

1. Graph

In your notebook, sketch and label the \(x\) and \(y\) coordinate axes, and the vector \(\vec{v}\).

2. Elaborate

Add to your sketch a right triangle that has \(\vec{v}\) as it’s hypotenuse.

3. Compute

Compute the magnitude of \(\vec{v}\).

4. Identify

Use what you know about isosceles right triangles to identify the angle \(\theta\) that \(\vec{v}\) makes with the positive \(x\)-axis.

Use that angle \(\theta\) and the unit circle to find the component form of the unit vector \(\hat{v}\) that has the same direction as \(\vec{v}\).

5. Graph & Estimate

Go to graph 1:

Identify the vector \(\vec{w} = \big< -4, 3 \big>\) (red).

At the top of the command window, locate the sliders labeled \(a\) and \(m\).

The \(a\)-slider controls the angle made by \(\vec{w}\) and the positive \(x\)-axis. The \(m\)-slider controls the magnitude \(| \vec{w} |\).

Use these sliders to adjust the magnitude and direction of the blue vector until it approximates \(\vec{w}\), and use it to estimate:

• The magnitude \(| \vec{w} |\).
• Estimate the components of the unit vector \(\hat{w}\) that has the same direction as \(\vec{w}\).
• The components of the unit vector whose direction is opposite to the direction of \(\vec{w}\).

BIG IDEAS

Equivalent Vectors & Standard Position (example)

Two vectors are equivalent if:

• They have the same magnitude & direction. (geometric)
• They have the same component form. (algebraic)

A vector is in standard position if its initial point is the origin \(O\).

Magnitude & Direction (exercise 1)

A unit vector \(\hat{v}\) is a vector whose magnitude equals 1:

\[| \hat{v} | = 1\]

Every unit vector \(\hat{v}\) in the plane ( \(\mathbb{R}^2\) ) has component form:

\[\hat{v} = \big< \cos \theta, \sin \theta \big>\]

If \(\vec{v} \neq \vec{0}\) is a non-zero vector, its direction vector is the unit vector \(\hat{v}\) that has the same direction as \(\vec{v}\).