Let u = [2,−3, 1] and v = [0, 1, 4]. Calculate the vector u − v. Place u and v tail-to-tail to form two sides of a triangle. With regard to this triangle, what does u−v represent?

Calculate the number u•u and discuss its relevance to the diagram you have drawn. Do the same for the number
(u − v) • (u − v).

Well, let's do some vector math and see what we get!

First, let's calculate u - v. When we subtract two vectors, we subtract their corresponding components. So, (u - v) = [2, -3, 1] - [0, 1, 4] = [2 - 0, -3 - 1, 1 - 4] = [2, -4, -3].

Now, when we place u and v tail-to-tail, we're essentially creating a triangle with u and v as two of its sides. The vector u - v represents the third side of this triangle.

As for u • u, that's the dot product of vector u with itself. We calculate it by multiplying the corresponding components and summing them up. So, (u • u) = (2*2) + (-3*-3) + (1*1) = 4 + 9 + 1 = 14.

In the context of the diagram, (u • u) represents the magnitude or length squared of vector u. It gives us information about the size of vector u.

Similarly, let's calculate (u - v) • (u - v). This is the dot product of vector u - v with itself. So, (u - v) • (u - v) = (2*2) + (-4*-4) + (-3*-3) = 4 + 16 + 9 = 29.

In the diagram, (u - v) • (u - v) represents the magnitude or length squared of vector u - v. It tells us about the size of the vector u - v.

So, to summarize, u - v represents the third side of the triangle formed by placing u and v tail-to-tail. (u • u) and (u - v) • (u - v) provide information about the magnitudes or lengths of vectors u and u - v, respectively.

To calculate the vector u - v, we subtract the corresponding components of u and v:

u - v = [2 - 0, -3 - 1, 1 - 4]
= [2, -4, -3]

By placing u and v tail-to-tail to form two sides of a triangle, the vector u - v represents the diagonal of the triangle, which connects the initial point of v to the terminal point of u.

To calculate the dot product u • u, we multiply the corresponding components of u and sum the results:
u • u = (2 * 2) + (-3 * -3) + (1 * 1)
= 4 + 9 + 1
= 14

The number u • u represents the magnitude or length squared of the vector u. In the triangle, it corresponds to the squared length of the side formed by vector u.

Similarly, to calculate the dot product (u - v) • (u - v), we multiply the corresponding components of (u - v) and sum the results:
(u - v) • (u - v) = (2 * 2) + (-4 * -4) + (-3 * -3)
= 4 + 16 + 9
= 29

The number (u - v) • (u - v) represents the magnitude or length squared of the vector (u - v). In the triangle, it corresponds to the squared length of the diagonal connecting the initial point of v to the terminal point of u.

To calculate the vector u - v, we subtract the corresponding components of the vectors.

u - v = [2, -3, 1] - [0, 1, 4]

u - v = [2 - 0, -3 - 1, 1 - 4]

u - v = [2, -4, -3]

Now, let's discuss what u - v represents with regard to the triangle formed by placing u and v tail-to-tail.

When we place u and v tail-to-tail, u - v represents the third side (or the resultant vector) of the triangle formed. It connects the end of vector v to the end of vector u, completing the triangle.

Now, let's calculate the dot product of u with itself, u • u:

u • u = (2 * 2) + (-3 * -3) + (1 * 1)
= 4 + 9 + 1
= 14

The dot product u • u is relevant to the diagram because it represents the magnitude (or length) squared of vector u. In the context of the triangle formed, it gives us the square of the length of vector u.

Similarly, let's calculate the dot product of (u - v) with itself, (u - v) • (u - v):

(u - v) • (u - v) = (2 * 2) + (-4 * -4) + (-3 * -3)
= 4 + 16 + 9
= 29

The dot product (u - v) • (u - v) is relevant to the diagram because it represents the magnitude (or length) squared of the resultant vector (u - v). In the context of the triangle formed, it gives us the square of the length of the third side of the triangle.

If you are studying vectors the above question cannot be made much easier.

I suggest just following the very explicit instructions.