Given the vectors v=-2i+5j and w=3i+4i, determine
(do I just plug what is given? but how would I find the length? is there a special way to solve vectors?)
1. 1/2 v
2. w-v
3. length of w
4. the unit vector for v
Use the pythogrean theorm to find length of w, i,j are at right angles to each other. The unit vector is the vector divided by length.
To solve the given vector problems, you'll need to manipulate the given vectors and apply some vector operations. Here's how you can find the answers:
1. To calculate 1/2 v, you can simply multiply each component of the vector v by 1/2. So, 1/2 v would be (-2/2)i + (5/2)j, which simplifies to -i + (5/2)j.
2. To find w-v, you subtract each component of vector v from the corresponding component of vector w. So, w-v would be (3-(-2))i + (4-5)j, which simplifies to 5i - 1j, or 5i - j.
3. To determine the length of a vector, you can use the Pythagorean theorem in a 2-dimensional space. The length of a vector w with components (w1, w2) is given by the formula length(w) = sqrt(w1^2 + w2^2). In this case, w = (3i + 4j), so the length of w would be length(w) = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5.
4. To find the unit vector for v, you need to divide the vector v by its length. First, calculate the length of v using the same formula as in step 3. Then, divide each component of v by the length of v to get the unit vector. So, the unit vector for v would be (-2/5)i + (5/5)j, which simplifies to -2/5i + j.
So, to summarize:
1. 1/2 v = -i + (5/2)j
2. w-v = 5i - j
3. length of w = 5
4. unit vector for v = -2/5i + j
Remember to always break down the problem into smaller steps, perform the necessary calculations, and simplify the result when needed.