8.26 2D Vectors: Components, Magnitude, and Addition

A set of practice flashcards focusing on vector components, magnitudes, directions, and vector addition in two dimensions, as discussed in the lecture notes.

What is the purpose of decomposing a vector into x and y components?

To determine how much of the vector lies along the x-axis (Ax) and y-axis (Ay), enabling analysis of motion and addition of vectors by their components.

What are Ax and Ay in a vector A?

Ax is the component along the x-axis; Ay is the component along the y-axis.

How do you add two vectors using their components?

Add their x-components to get Cx and their y-components to get Cy; the resultant vector C has components (Cx, Cy).

What is the geometric meaning of a vector’s magnitude and direction?

Magnitude is the length of the vector; direction is the angle it makes with a reference axis.

What is the magnitude formula for a vector with components Ax and Ay?

|A| = sqrt(Ax^2 + Ay^2).

How do you find the direction angle theta from Ax and Ay?

Compute theta = arctan(|Ay|/|Ax|) and determine the quadrant from the signs of Ax and Ay.

Which angle should you use to find components, the one that touches the x-axis or the one that touches the y-axis, and why?

Use the angle that touches the x-axis so that Ax = |A| cos(theta) and Ay = |A| sin(theta).

If a vector has length 5 m and makes a 37-degree angle with the +x axis in the first quadrant, what are its components?

Ax ≈ 5 cos(37°) ≈ 3.9 m; Ay ≈ 5 sin(37°) ≈ 3.0 m.

What is the 3-4-5 triangle used for in vector problems?

It provides easy side lengths that yield common angles (37° and 53°) to simplify cosine and sine values.

How can you verbally describe a vector’s direction in a 2D plane?

Use phrases like '37° to the right of north' or 'east of north' to specify the direction.

What does the overbar on a symbol denote in vector notation?

The quantity is a vector, which has both magnitude and direction.

What are cx and cy in terms of a and b when adding vectors A and B?

cx = ax + bx and cy = ay + by.

How do you combine three vectors A, B, C into D in terms of components?

Dx = Ax + Bx + Cx; Dy = Ay + By + Cy; magnitude |D| = sqrt(Dx^2 + Dy^2).

What does the 'shadow' concept mean in vector components?

A vector’s component along an axis is its projection onto that axis—the shadow it casts on that axis.

What is the 'tip-to-tail' rule for vector addition?

Place the tail of the second vector at the tip of the first; the resultant vector goes from the start of the first to the end of the second.

What is a quick radian vs degree check using sin(90)?

If sin(90) evaluates to 1, you’re in degree mode; if not, you’re in radian mode (check and switch as needed).

How do you determine the sign of vector components when projecting onto axes?

Use the vector’s quadrant to assign signs to Ax and Ay (e.g., negative x and positive y in quadrant II).

What is a practical method to keep track of multiple vectors and their sum?

Create a component table listing each vector’s x and y components, then sum the columns to get the resultant components.

What does |D| represent once Dx and Dy are known, and how do you compute it?

|D| is the magnitude (length) of D, computed as sqrt(Dx^2 + Dy^2), independent of direction.