Compute the following, then sketch each resultant vector.
𝘂 = 5𝗶 − 𝗷, 𝘃 = −2𝗶 + 𝗷
a) u+v
b) u-v
c) 2u − 3v
d) 1/2v + 4u
Just do the simple arithmetic.
I will do c)
2u - 3v
= 2(5𝗶 − 𝗷) - 3(−2𝗶 + 𝗷)
= 10i - 2j + 6i - 3j
= 16i - 5j
I will assume you know how to sketch that resultant vector. (16 to the right, 5 down)
To compute each resultant vector, we will use vector addition and scalar multiplication. Let's go step by step.
Given:
𝘂 = 5𝗶 − 𝗷
𝘃 = −2𝗶 + 𝗷
a) To compute u + v, we add the corresponding components of 𝘂 and 𝘃:
𝘂 + 𝘃 = (5𝗶 − 𝗷) + (−2𝗶 + 𝗷)
= 5𝗶 − 𝗷 − 2𝗶 + 𝗷
= 5𝗶 − 2𝗶 − 𝗷 + 𝗷
= 3𝗶
So, 𝘂 + 𝘃 = 3𝗶.
The resultant vector, 3𝗶, is a vector in the positive x-direction.
b) To compute u - v, we subtract the corresponding components of 𝘂 and 𝘃:
𝘂 − 𝘃 = (5𝗶 − 𝗷) − (−2𝗶 + 𝗷)
= 5𝗶 − 𝗷 + 2𝗶 − 𝗷
= 5𝗶 + 2𝗶 − 𝗷 − 𝗷
= 7𝗶 − 2𝗷
So, 𝘂 - 𝘃 = 7𝗶 − 2𝗷.
The resultant vector, 7𝗶 - 2𝗷, has a component in the positive x-direction and a component in the negative y-direction.
c) To compute 2𝘂 − 3𝘃, we multiply each component of 𝘂 by 2 and each component of 𝘃 by -3, then add the corresponding components:
2𝘂 − 3𝘃 = 2(5𝗶 − 𝗷) − 3(−2𝗶 + 𝗷)
= 10𝗶 − 2𝗷 + 6𝗶 − 3𝗷
= 16𝗶 − 5𝗷
So, 2𝘂 − 3𝘃 = 16𝗶 − 5𝗷.
The resultant vector, 16𝗶 - 5𝗷, has a component in the positive x-direction and a component in the negative y-direction.
d) To compute 1/2𝘃 + 4𝘂, we multiply each component of 𝘃 by 1/2 and each component of 𝘂 by 4, then add the corresponding components:
1/2𝘃 + 4𝘂 = 1/2(−2𝗶 + 𝗷) + 4(5𝗶 − 𝗷)
= -𝗶 + 1/2𝗷 + 20𝗶 − 4𝗷
= 19𝗶 - 7/2𝗷
So, 1/2𝘃 + 4𝘂 = 19𝗶 - 7/2𝗷.
The resultant vector, 19𝗶 - 7/2𝗷, has a component in the positive x-direction and a component in the negative y-direction.
Now, let's sketch each resultant vector:
a) 𝘂 + 𝘃 = 3𝗶. This vector points only in the positive x-direction.
b) 𝘂 - 𝘃 = 7𝗶 - 2𝗷. This vector has a component in the positive x-direction and a component in the negative y-direction.
c) 2𝘂 − 3𝘃 = 16𝗶 − 5𝗷. This vector has a component in the positive x-direction and a component in the negative y-direction.
d) 1/2𝘃 + 4𝘂 = 19𝗶 - 7/2𝗷. This vector has a component in the positive x-direction and a component in the negative y-direction.
Please note that the sketches should represent the direction and relative size of the vectors.
To compute the resultant vectors u+v, u-v, 2u - 3v, and 1/2v + 4u, we need to perform vector addition and scalar multiplication.
First, let's find the resultant vector u+v:
To compute u+v, we simply add the corresponding components.
u+v = (5𝗶 − 𝗷) + (−2𝗶 + 𝗷)
= 5𝗶 − 𝗷 − 2𝗶 + 𝗷
= (5𝗶 − 2𝗶) + (−𝗷 + 𝗷)
= 3𝗶 + 0𝗷
= 3𝗶
The resultant vector u+v is 3𝗶. To sketch this, imagine a vector with magnitude 3 in the positive x-direction.
Next, let's find the resultant vector u-v:
To compute u-v, we subtract the corresponding components.
u-v = (5𝗶 − 𝗷) - (−2𝗶 + 𝗷)
= 5𝗶 − 𝗷 + 2𝗶 - 𝗷
= (5𝗶 + 2𝗶) + (−𝗷 - 𝗷)
= 7𝗶 - 2𝗷
The resultant vector u-v is 7𝗶 - 2𝗷. To sketch this, imagine a vector with magnitude 7 in the positive x-direction and a vector with magnitude 2 in the negative y-direction.
Now, let's find the resultant vector 2u − 3v:
To compute 2u − 3v, we multiply each component by the scalar and then perform vector subtraction.
2u − 3v = 2(5𝗶 − 𝗷) − 3(−2𝗶 + 𝗷)
= 10𝗶 − 2𝗷 + 6𝗶 - 3𝗷
= (10𝗶 + 6𝗶) + (−2𝗷 - 3𝗷)
= 16𝗶 - 5𝗷
The resultant vector 2u − 3v is 16𝗶 - 5𝗷. To sketch this, imagine a vector with magnitude 16 in the positive x-direction and a vector with magnitude 5 in the negative y-direction.
Lastly, let's find the resultant vector 1/2v + 4u:
To compute 1/2v + 4u, we multiply each component by the scalar and then perform vector addition.
1/2v + 4u = 1/2(−2𝗶 + 𝗷) + 4(5𝗶 − 𝗷)
= -𝗶/1 + 1/2𝗷 + 20𝗶 - 4𝗷
= -𝗶/1 + 20𝗶 + 1/2𝗷 - 4𝗷
= (-1/1 + 20)𝗶 + (1/2 - 4)𝗷
= 19𝗶 - 3.5𝗷
The resultant vector 1/2v + 4u is 19𝗶 - 3.5𝗷. To sketch this, imagine a vector with magnitude 19 in the positive x-direction and a vector with magnitude 3.5 in the negative y-direction.
To summarize, the resultant vectors are:
a) u+v = 3𝗶
b) u-v = 7𝗶 - 2𝗷
c) 2u - 3v = 16𝗶 - 5𝗷
d) 1/2v + 4u = 19𝗶 - 3.5𝗷