(II) Figure 36 shows two vectors, 𝐀⃗ and 𝐁⃗, whose magni- tudes are A=6.8 units

(II) Figure 36 shows two vectors, 𝐀⃗ and 𝐁⃗, whose...

$$\begin{array}{l}{\text { (II) Figure } 36 \text { shows two vectors, } \vec{\mathbf{A}} \text { and } \vec{\mathbf{B}}, \text { whose magni- }} \\ {\text { tudes are } A=6.8 \text { units and } B=5.5 \text { units. Determine } \vec{\mathbf{C}} \text { if }}\end{array} $$ $$ \begin{array}{l}{\text { (a) } \mathbf{C}=\mathbf{A}+\mathbf{B},(b) \mathbf{C}=\mathbf{A}-\mathbf{B},(c) \mathbf{C}=\mathbf{B}-\mathbf{A} \text { . Give the }} \\ {\text { magnitude and direction for cach. }}\end{array} $$

Key Concepts

Direction (Angle) Calculation

Calculating the direction of a vector involves determining the angle it makes with a reference axis, such as the horizontal axis in a two-dimensional plane. This is commonly done using trigonometric functions like the arctan function to compute the angle from the vector's components. Understanding the direction is important for interpreting the orientation of the vector in space.

Magnitude Calculation

The magnitude of a vector is a measure of its length or size and represents the distance from the tail to the head of the vector. In Cartesian coordinates, it is typically calculated using the Pythagorean theorem, which involves taking the square root of the sum of the squares of its components. This concept is fundamental in quantifying vectors in physics and engineering.

Vector Addition

Vector addition is the process of combining two or more vectors to produce a resultant vector. It involves adding corresponding components of the vectors or using graphical methods such as the parallelogram or triangle rule. This concept is essential because it helps determine the net effect when multiple forces or displacements are applied simultaneously.

Vector Subtraction

Vector subtraction involves finding the difference between two vectors, which represents the relative change or displacement from one vector to another. It is conceptually similar to addition but involves reversing the direction of one vector before performing the summation. This operation is crucial in many applications, such as determining relative velocities or changes in position.

Transcript

00:01 Our question says, figure 36 shows two vectors, vector a and b, whose magnitudes are a equals 6 .8 units, and b equals 5 .5 units.

00:10 Determined c if c is, if a, c is equal to a plus b, b, b, c is equal to a minus b, and c, c is equal to b minus a.

00:20 Give the magnitude and direction for each of the situations.

00:23 So i redrew figure 36, which shows the x and y coordinate system, and it shows vector a going in the positive x direction, and vector b going in the negative x direction along the x axis.

00:32 So i wrote down here that vector a is equal to 6 .8 units, which i write as u, in the positive x hat direction, since it's moving, since the hat represents directionality of a coordinate system.

00:45 Then b is equal to 5 .5 units, but it's in the minus x hat direction.

00:50 Okay? so then the question is, what is c equal for the following situations? and i wrote the situations we were given in red.

00:56 And i'm going to go ahead and do the calculations in black.

01:00 Okay.

01:01 So if c is equal to a plus b, we'll write out what we have for a, 6 .8 units, right? and that's in the positive x hat direction, plus, and then b.

01:15 Well, we can pull that negative sign out front, okay, from the negative x had direction.

01:22 And that is equal to 5 .5 units.

01:24 And since we pulled the negative sign out front, we just write this as x hat.

01:29 Okay.

01:31 Well, 6 .8u plus negative 5 .5u, we can add them together, since they're both in the x -hat direction, plus a negative is the same as minus that number.

01:40 The difference between those two is 1 .3 units.

01:44 And direction plus x -hat in the positive x -hat direction.

01:52 Fox out in as a solution to part a.

01:54 Part b, we want a minus b...

Please give Ace some feedback