For which value(s) of k will the dot product of the vectors (k,2k-1, 3) and (k,5,-4) be 7?

I did this so far,

2k-1=5
6/2

k=3

what's the problem?

(k,2k-1,3)•(k,5,-4) = k*k + (2k-1)(5) + 3(-4)

so, we need

k^2 + 10k - 5 - 12 = 7
k^2 + 10k - 24 = 0
(k-2)(k+12) = 0
so, k=2 or -12

k=2: (2,3,3)•(2,5,-4) = 4+15-12 = 7
k=-12: (-12,-25,3)•(-12,5,-4) = 144-125-12 = 7

To find the value(s) of k that will result in a dot product of 7 between the given vectors, let's go step by step.

The dot product of two vectors is calculated by multiplying the corresponding components of the vectors and then summing up those products.

The given vectors are (k, 2k-1, 3) and (k, 5, -4).

To calculate the dot product, we multiply the corresponding components and sum them up:

(k * k) + ((2k-1) * 5) + (3 * -4)

Expanding this equation, we get:

k^2 + 10k - 5 + -12

Combining like terms, we have:

k^2 + 10k - 17

We want this expression to result in 7, so we set the equation equal to 7:

k^2 + 10k - 17 = 7

Rearranging the equation, we get:

k^2 + 10k - 24 = 0

Now we need to solve this quadratic equation for k. Factoring may not be straightforward in this case, so we can use the quadratic formula:

k = (-b ± √(b^2 - 4ac)) / 2a

In our equation, a = 1, b = 10, and c = -24. Substituting these values into the formula, we have:

k = (-10 ± √(10^2 - 4(1)(-24))) / (2(1))

Simplifying further:

k = (-10 ± √(100 + 96)) / 2

k = (-10 ± √196) / 2

k = (-10 ± 14) / 2

We have two possible solutions:

1) If k = (-10 + 14) / 2:

k = 4 / 2

k = 2

2) If k = (-10 - 14) / 2:

k = -24 / 2

k = -12

Therefore, the dot product of the given vectors will be 7 for the values k = 2 and k = -12.

To find the value(s) of k for which the dot product of the given vectors is 7, we need to use the definition of the dot product and set it equal to 7.

The dot product of two vectors A = (a1, a2, a3) and B = (b1, b2, b3) is given by:
A · B = a1 * b1 + a2 * b2 + a3 * b3

In this case, the vectors are
A = (k, 2k - 1, 3) and B = (k, 5, -4).

Setting up the dot product equation:
(k)(k) + (2k - 1)(5) + (3)(-4) = 7

Simplifying the equation:
k^2 + 10k - 5 + (-12) = 7
k^2 + 10k - 5 - 12 = 7
k^2 + 10k - 17 = 7

Rearranging the equation:
k^2 + 10k - 24 = 0

To solve this quadratic equation, we can either factor it or use the quadratic formula. Let's use the quadratic formula:
k = (-b ± sqrt(b^2 - 4ac)) / 2a

For our quadratic equation k^2 + 10k - 24 = 0, a = 1, b = 10, and c = -24.

Plugging these values into the quadratic formula:
k = (-10 ± sqrt(10^2 - 4(1)(-24))) / 2(1)
k = (-10 ± sqrt(100 + 96)) / 2
k = (-10 ± sqrt(196)) / 2
k = (-10 ± 14) / 2

Now we have two possible values for k:
k1 = (-10 + 14) / 2 = 2/2 = 1
k2 = (-10 - 14) / 2 = -24/2 = -12

So, the dot product of the given vectors (k, 2k - 1, 3) and (k, 5, -4) is equal to 7 for k = 1 and k = -12.