![Picture](/uploads/4/3/9/7/43974927/3875650_orig.png)
Vector: A quantity that has both direction and magnitude(length)
Vector Components: All vectors can be broken up into their x- and y-components to "simplify" things.
The vertical components is the y-component, and the horizontal is the x. If the original vector is V1, the components can be denoted as V1x and V1y. If you know the vector is at a certain degrees, then the components can be found using trigonometry. If the angle between the vector and the horizontal component in the bottom left of the picture is angle a, then V1 cos a = V1x and V1 sin a=V1y. Similarly, if you are given the components of a vector, that vector's magnitude can be found using the Pythagorean theorem and its direction found using the inverse tangent.
![Picture](/uploads/4/3/9/7/43974927/8261373.jpg?375)
Problem Example: Find the x- and y-components of the following vectors.
a. A vector of magnitude 7, 15 degrees above the x-axis
7cos15=Vx
Vx=6.8
7sin15=Vy
Vy=1.8
b. A vector of magnitude 11, 20 degrees above the x-axis
11cos20=Vx
Vx=10.3
11sin20=Vy
Vy=3.76
Find the magnitude and direction of the vector with the following components:
c. Vx=3
Vy=4
3^2+4^2=V^2
V=5
tangent of angle theta = 4/3
angle theta=inverse tangent4/3
angle theta=53 degrees
magnitude=5
direction=53 degrees
a. A vector of magnitude 7, 15 degrees above the x-axis
7cos15=Vx
Vx=6.8
7sin15=Vy
Vy=1.8
b. A vector of magnitude 11, 20 degrees above the x-axis
11cos20=Vx
Vx=10.3
11sin20=Vy
Vy=3.76
Find the magnitude and direction of the vector with the following components:
c. Vx=3
Vy=4
3^2+4^2=V^2
V=5
tangent of angle theta = 4/3
angle theta=inverse tangent4/3
angle theta=53 degrees
magnitude=5
direction=53 degrees
Adding Vectors:
This video shows how to add vectors using the tail-to-tip method. The parallelogram method may also be used. When you add vectors, you end up with what is called a resultant vector. you can also use vector components to add vectors. When you add the x-component of one vector to another, you get the x-component of the resultant vector. It is the same for the y-components. Once you have calculated the components for the resultant vector, you can find both that vector's magnitude and direction (length and angle).
Subtracting works in a similar way to adding vectors, except when drawing them, the vector you want to subtract should be drawn in the opposite direction than how you would draw it when adding. Think of a negative vector as the same vector but with an arrow on the opposite end. Then you can take the same approach as adding vectors. This opposite thing works because if you are subtracting vector Y from vector V, it is the same thing as V+(-Y). If you want to do it by components, just subtract the components you find instead of adding them.
Vector Word Problem:
Keita left camp three days ago on a journey into the jungle. The three days of his journey can be described by displacement (distance and direction) vectors d1⃗, d2⃗, and d3⃗.
d1⃗=(7,8)
d2⃗=(6,2)
d3⃗=(2,9) (Distances are given in kilometers, km.)
How far is Keita from camp at the end of day three?(Round your final answer to the nearest tenth.) 24.2km
What direction is Keita from camp at the end of day three?(Round your final answer to the nearest degree. Your answer should be between 0 and 180∘.) 52 degrees
Keita left camp three days ago on a journey into the jungle. The three days of his journey can be described by displacement (distance and direction) vectors d1⃗, d2⃗, and d3⃗.
d1⃗=(7,8)
d2⃗=(6,2)
d3⃗=(2,9) (Distances are given in kilometers, km.)
How far is Keita from camp at the end of day three?(Round your final answer to the nearest tenth.) 24.2km
What direction is Keita from camp at the end of day three?(Round your final answer to the nearest degree. Your answer should be between 0 and 180∘.) 52 degrees
First, looking at the distance, we want the sum of all three of the vectors given. The horizontal and vertical components of these vectors have been given (in the parentheses), so all we need to do is add them. d1 has components (7,8), d2 has (6,2), and d3 has (2,9), so the components of our resultant vector, or the vector that shows the total distance traveled, has the components (7+6+2, 8+2+9), or (15,19). We can then use these components in the Pythagorean theorem to find the total distance of all three vectors. 15^2+19^2=magnitude^2, 225+361=magnitude^2, 586=magnitude^2, magnitude=24.2km. Now we need to find the direction, which is going to be the angle in the bottom left if we imagined the horizontal line directly under the first vector to be the x-axis. So that angle is going to be θ. Now we use the vector components (15,19) to find θ. Tan θ= (19/15), θ=tan^-1 (19/15), which is 52 degrees.