Algebraic Representation and Components

Breaking Down the Vector

In physics, diagonal movement is often complex to calculate. We simplify this by resolving a vector into its horizontal and vertical components.

Think of a vector not as one single arrow, but as the result of moving along the x-axis and then the y-axis.

Welcome to the world of vector components. Imagine this diagonal vector represents the velocity of a plane. To make our math easier, we don't look at the diagonal directly. Instead, we look at how fast it moves horizontally, and how fast it moves vertically. Together, these two perpendicular parts make up the whole.

• A vector can be broken into perpendicular components.
• The original vector acts as the hypotenuse of a right triangle.
• Components simplify mathematical operations in physics.

The Math of Resolution

Using trigonometry, we can calculate the exact length of the components if we know the magnitude (A) and the angle (θ).

• Ax = A cos(θ)
• Ay = A sin(θ)

How do we find the exact values? We use trigonometry. When the angle theta is measured from the x-axis, the horizontal component Ax is the adjacent side, so we use cosine. The vertical component Ay is the opposite side, so we use sine. Let's see how changing the angle changes these values. Notice as the angle increases, the vertical component grows while the horizontal component shrinks.

• Cosine relates to the adjacent side (x-component).
• Sine relates to the opposite side (y-component).
• The angle must be measured from the positive x-axis.

Unit Vector Notation

Standard unit vectors i, j, and k are the building blocks of algebraic vector notation. They have a magnitude of 1 and point along the axes.

• i: x-direction
• j: y-direction
• k: z-direction (3D)

To write vectors algebraically, we use unit vectors. Think of 'i' and 'j' as directional pointers. The unit vector 'i' always points one unit in the x-direction. 'j' points one unit in the y-direction. So, a vector that moves 4 units right and 3 units up is simply written as 4i plus 3j.

• Unit vectors are dimensionless and have a magnitude of 1.
• Vector V = Vx i + Vy j + Vz k.
• This notation makes adding and subtracting vectors simple.

Reconstructing the Vector

What if you have the components but need the total magnitude and direction? We reverse the process.

• Magnitude: Use the Pythagorean theorem.
• Direction: Use the inverse tangent function.

Now, let's try the reverse. Enter values for the x and y components. We use the Pythagorean theorem to find the magnitude, or the length of the arrow. Then, we use the inverse tangent of the vertical over the horizontal to find the angle. Try entering a negative x value and see what happens to the angle.

• |V| = sqrt(Vx² + Vy²)
• θ = tan⁻¹(Vy / Vx)
• Always check your quadrant!

Physics in Action: The Soccer Kick

A soccer ball is kicked at 20 m/s at an angle of 30°.

Practice resolving this into velocity components to see how fast it travels along the ground versus into the air.

Let's apply this to physics. We kick a ball at 20 meters per second at a 30-degree angle. Using cosine, we find it moves forward at 17.3 meters per second. Using sine, it rises at 10 meters per second. In algebraic form, we write this as 17.3 i plus 10 j.

• Horizontal velocity determines distance.
• Vertical velocity determines height and time in air.
• V = (17.3i + 10.0j) m/s

Diagnostic: The Tangent Trap

A vector has components Vx = -5 and Vy = 5. Your calculator says the angle is -45°. Is this correct? Explain your reasoning.

Time for a logic check. Look at the components: x is negative 5, y is positive 5. The calculator gives you negative 45 degrees. Type your diagnosis: is that angle correct for the quadrant we are in? Why or why not?

• Calculators only return tan⁻¹ values for Quadrants 1 and 4.
• Vectors in Quadrant 2 and 3 require a 180° adjustment.