Find the component form of v and sketch the specified vector operations geometrically, where u = 4i − j, and w = i + 5j.
v = −u + w
v = ( , )
v = -(4i-j) + i+5j
= (-4,1) + (1,5)
= (-3,6)
To find the component form of v, we need to subtract u from w.
Given that u = 4i - j and w = i + 5j, we can write the vector v as:
v = -u + w
To simplify, we distribute the negative sign to u and add it to w:
v = -u + w
v = -4i + j + i + 5j
v = -4i + i + j + 5j
v = -3i + 6j
So, the component form of v is (-3, 6).
To find the component form of v, we need to subtract u from w:
v = w - u
First, let's find the component form of w:
w = i + 5j
So, the x-component of w is 1 and the y-component is 5.
Now, let's find the component form of u:
u = 4i - j
So, the x-component of u is 4 and the y-component is -1.
Now, let's subtract u from w to find the component form of v:
v = (1 - 4)i + (5 - (-1))j
v = -3i + 6j
Therefore, the component form of v is (-3, 6).
To sketch the specified vector operations geometrically, follow these steps:
1. Draw a coordinate system with two perpendicular axes (x and y).
2. Start from the origin (0, 0).
3. Draw a vector w with a component form of (1, 5) by moving 1 unit in the x-direction and 5 units in the y-direction from the origin.
4. Draw a vector u with a component form of (4, -1) by moving 4 units in the x-direction and -1 unit in the y-direction from the origin.
5. To sketch v = -u + w, start at the endpoint of vector u and move in the opposite direction of u (in this case, to the left) by 4 units in the x-direction and -1 unit in the y-direction.
6. Then, from that new point, move in the same direction as vector w (in this case, to the right) by 1 unit in the x-direction and 5 units in the y-direction.
7. The endpoint of the vector v will be the point where the final displacement ends.
8. Finally, draw a straight line from the origin to the endpoint of vector v to represent the vector v geometrically.
Remember to label each vector with their respective names and indicate the length and direction of each vector.