Determine all values of k for which each pair of vectors is orthogonal.
a) (1,2) and (k,k)
b) (1,2,1) and (k,2k,4)
a) To determine if two vectors are orthogonal, we need their dot product to be equal to zero. The dot product of two vectors (x1, y1) and (x2, y2) is given by: x1x2 + y1y2.
For the vectors (1,2) and (k,k), the dot product is given by: 1*k + 2*k = k + 2k = 3k.
For these vectors to be orthogonal, their dot product must be equal to zero. Therefore, we have the equation 3k = 0. Solving for k, we find that k = 0.
So, for the vectors (1,2) and (k,k) to be orthogonal, k must equal 0.
b) For the vectors (1,2,1) and (k,2k,4), the dot product is given by: 1*k + 2*2k + 1*4 = k + 4k + 4 = 5k + 4.
For these vectors to be orthogonal, their dot product must be equal to zero. Therefore, we have the equation 5k + 4 = 0. Solving for k, we find that k = -4/5.
So, for the vectors (1,2,1) and (k,2k,4) to be orthogonal, k must equal -4/5.
a) To determine if two vectors are orthogonal, their dot product must equal zero.
The dot product of (1,2) and (k,k) is given by
(1)(k) + (2)(k) = k + 2k = 3k.
For this dot product to equal zero, we set 3k = 0 and solve for k.
3k = 0
k = 0.
Therefore, the value of k for which (1,2) and (k,k) are orthogonal is k = 0.
b) Similarly, for vectors (1,2,1) and (k,2k,4), their dot product is given by
(1)(k) + (2)(2k) + (1)(4) = k + 4k + 4 = 5k + 4.
For these vectors to be orthogonal, we set 5k + 4 = 0 and solve for k.
5k + 4 = 0
5k = -4
k = -4/5.
Therefore, the value of k for which (1,2,1) and (k,2k,4) are orthogonal is k = -4/5.
To determine the values of k for which each pair of vectors is orthogonal, we need to check if the dot product of the vectors is zero. The dot product of two vectors is calculated by multiplying corresponding components of the vectors and summing them up.
a) For the vectors (1,2) and (k,k), the dot product is given by:
(1)(k) + (2)(k) = k + 2k = 3k
For the vectors to be orthogonal, the dot product should be zero. Therefore, the equation is:
3k = 0
To solve for k, we divide both sides of the equation by 3:
k = 0
So, the value of k that makes the vectors orthogonal is k = 0.
b) For the vectors (1,2,1) and (k,2k,4), the dot product is given by:
(1)(k) + (2)(2k) + (1)(4) = k + 4k + 4 = 5k + 4
For the vectors to be orthogonal, the dot product should be zero. Therefore, the equation is:
5k + 4 = 0
To solve for k, we subtract 4 from both sides of the equation:
5k = -4
Then, divide both sides of the equation by 5:
k = -4/5
So, the value of k that makes the vectors orthogonal is k = -4/5.