Find a vector that has the same direction as the given vector but has length 6.

‹-2, -2, 4›

I know how to find a unit vector of this, but to find a vector in the same direction but length >1 I'm not sure how to determine the scalar multiple that will be needed. I know it will be >1 because the length of the current vector is sqrt(24) and l=6 would be l=sqrt(36).

I tried using the multiple of sqrt(36/24) but it didn;t work.

Any help would be appreciated.'

a vector with the same direction is (-1,-1,2)

a unit vector in that same direction is
1/√6(-1,-1,2) or (-1/√6, -1/√6, 2/√6)

so a vector 6 times that long is (-6/√6, -6/√6, 12/√6)
or
(-√6 , -√6 , 2√6) after rationalizing the denominator.

To find a vector with the same direction as the given vector but with a different length, you can follow these steps:

1. Calculate the length (magnitude) of the given vector. Let's call it "L1".
The length of a vector ‹a, b, c› can be found using the formula:
L1 = √(a^2 + b^2 + c^2)

In this case, the given vector is ‹-2, -2, 4›:
L1 = √((-2)^2 + (-2)^2 + 4^2) = √(4 + 4 + 16) = √24

2. Determine the desired length for the new vector. Let's call it "L2".
In this case, you want a length of 6:
L2 = 6

3. Calculate the scalar multiplier (k) required to achieve the desired length.
The scalar multiplier (k) can be calculated using the formula:
k = L2 / L1

In this case:
k = 6 / √24

4. Multiply each component of the given vector by the scalar multiplier (k).
The resulting vector will have the same direction but will have a different length.

Using the given vector ‹-2, -2, 4› and the calculated scalar multiplier (k),
the new vector ‹a, b, c› can be found as:
a = -2 * (6 / √24)
b = -2 * (6 / √24)
c = 4 * (6 / √24)

Hence, the vector with the same direction as ‹-2, -2, 4› but with a length of 6 is approximately ‹-2.12, -2.12, 4.24› (rounded to two decimal places).