Question

Let u = (1, 2, 3), v = (2, 2, -1), and w = (4, 0, −4). Find z, where 2u + v - w+ 3z = 0. z = (No Response)

Answer 1

To find z in the equation 2u + v - w + 3z = 0, substitute the values and solve for z.

Step-by-step explanation:

To find the value of z in the equation 2u + v - w + 3z = 0, we can substitute the known values for u, v, and w and solve for z. First, let's substitute the values: 2u + v - w + 3z = 0 becomes 2(1, 2, 3) + (2, 2, -1) - (4, 0, -4) + 3z = 0. Simplifying, we get (2, 4, 6) + (2, 2, -1) - (4, 0, -4) + 3z = 0.

Adding corresponding components, we have (2+2-4, 4+2, 6-1+4) + 3z = 0, which simplifies to (0, 6, 9) + 3z = 0.

To solve for z, we subtract (0, 6, 9) from both sides, resulting in 3z = -(0, 6, 9). Dividing by 3, we find that z = -(0, 2, 3).

Answer 2

To find the vector z such that 2u + v - w + 3z = 0, perform vector addition with the given vectors after scaling and negation, then solve for z by dividing the resultant vector by 3. The resulting vector z is (0, 2, 3).

Step-by-step explanation:

To find vector z, where 2u + v - w + 3z = 0, first calculate the sum of the vectors 2u, v, and -w. Let's perform the vector addition first:

• Let u = (1, 2, 3), therefore 2u = (2, 4, 6).
• Let v = (2, 2, -1).
• Let w = (4, 0, -4), therefore -w = (-4, 0, 4).

Now, add these vectors together:

• 2u + v = (2+2, 4+2, 6-1) = (4, 6, 5).
• Then, 2u + v - w = (4-4, 6+0, 5+4) = (0, 6, 9).

Substitute 2u + v - w into the equation:

0 = - (2u + v - w) + 3z

Therefore, 3z = (2u + v - w)

And z = (1/3)*(2u + v - w)

So, z would be z = (1/3)*(0, 6, 9) = (0, 2, 3).

The vector z equals (0, 2, 3) in this case.