Vector V⃗ 1 is 6.5 units long and points along the negative x axis. Vector V⃗ 2 is 8.3 units long and points at 60 ∘ to the positive x axis.
To solve this problem, let's break it down into steps:
Step 1: Draw a coordinate system. Place the origin (0,0) at the center of the coordinate system.
Step 2: Draw vector V⃗ 1. Since V⃗ 1 points along the negative x-axis, it will have a negative x component and a y component of 0. The length of V⃗ 1 is 6.5 units, so draw a line segment from the origin that extends 6.5 units in the negative x direction.
Step 3: Draw vector V⃗ 2. V⃗ 2 points at 60∘ to the positive x-axis. To find the components of V⃗ 2, we can use the trigonometric relationships:
x component = length * cos(angle)
y component = length * sin(angle)
In this case, the length of V⃗ 2 is 8.3 units and the angle is 60∘. Plugging the values into the equations, we have:
x component = 8.3 * cos(60∘) ≈ 4.15
y component = 8.3 * sin(60∘) ≈ 7.19
Draw a line segment from the origin that extends 4.15 units in the positive x direction and 7.19 units in the positive y direction.
Step 4: Label the vectors. Label the first vector as V⃗ 1 and the second vector as V⃗ 2.
That's it! You have successfully drawn vector V⃗ 1 and V⃗ 2.
To find the angle between the two vectors, we can use the dot product formula:
V⃗ 1 · V⃗ 2 = ||V⃗ 1|| ||V⃗ 2|| cos(θ)
where V⃗ 1 · V⃗ 2 is the dot product of the two vectors, ||V⃗ 1|| and ||V⃗ 2|| are the lengths of the vectors, and θ is the angle between them.
First, let's find the dot product:
V⃗ 1 · V⃗ 2 = ||V⃗ 1|| ||V⃗ 2|| cos(θ) = 6.5 * 8.3 * cos(θ)
Since V⃗ 1 points along the negative x-axis, its angle with the positive x-axis is 180 degrees. And since V⃗ 2 points 60 degrees to the positive x-axis, its angle with the positive x-axis is 60 degrees.
Using the dot product formula, we can substitute the values:
6.5 * 8.3 * cos(θ) = 6.5 * 8.3 * cos(180 - 60)
Simplifying this equation:
6.5 * 8.3 * cos(θ) = 6.5 * 8.3 * cos(120)
Now, we can cancel out the lengths:
cos(θ) = cos(120)
Using a calculator or trigonometric identity, we find that cos(120) = -0.5.
So, we have:
-0.5 = cos(θ)
To find θ, take the inverse cosine (arccos) of both sides:
θ = arccos(-0.5)
Using a calculator, we find that θ ≈ 120 degrees.
Therefore, the angle between the two vectors, V⃗ 1 and V⃗ 2, is approximately 120 degrees.
Vector V1:
Vx = -6.5 units.
Vy = 0.
Vector V2:
V = 8.3[60o]
Vx = 8.3*Cos60 =
Vy = 8.3in60 =
V1 + V2 = -6.5 + 8.3[60] =
-6.5 + 4.15+7.19i = -2.35 + 7.19i
= 7.56[71.9o] N. of W. = 108o CCW