Determine the value(s) of k such that the angle between the vectors a= (1,1,k) and b=(1,0,1) is 45 degrees.
a dot b = |a| |b|cosØ
(1,1,k) dot (1,0,1) = √(1+1+k^2) √(1+0+1)cos45°
1 + 0 + k = √(k^2 + 2)√2 * 1/√2
1+k = √(k^2+2)
square both sides
1 + 2k + k^2 = k^2 + 2
2k = 1
k = 1/2
check my arithmetic
Well, let's put on our funny glasses and solve this problem with a smile, shall we?
To find the angle between two vectors, we can use the dot product formula. The dot product of vectors a and b is given by:
a · b = |a| |b| cos(θ)
Where θ is the angle between the vectors. In this case, we want the angle to be 45 degrees, so let's plug in the values:
(1, 1, k) · (1, 0, 1) = sqrt(3) * sqrt(2) * cos(45)
Taking the dot product and simplifying, we get:
1 + k = sqrt(6) * cos(45)
Now, cos(45) is equal to 1/sqrt(2), so we can substitute that in:
1 + k = sqrt(6) * 1/sqrt(2)
Simplifying a bit further:
1 + k = sqrt(3)
Subtracting 1 from both sides, we find:
k = sqrt(3) - 1
And there you have it! The value of k that makes the angle between the vectors a = (1, 1, k) and b = (1, 0, 1) exactly 45 degrees is k = sqrt(3) - 1. Stay funny, my friend!
To find the value(s) of k for which the angle between the vectors a = (1, 1, k) and b = (1, 0, 1) is 45 degrees, we can use the dot product formula for finding the angle between two vectors.
The dot product of two vectors a and b is given by:
a · b = |a| |b| cos(theta)
where a · b is the dot product, |a| and |b| are the magnitudes of vectors a and b, and theta is the angle between the vectors.
First, let's calculate the magnitudes of the vectors a and b:
|a| = sqrt(1^2 + 1^2 + k^2) = sqrt(2 + k^2)
|b| = sqrt(1^2 + 0^2 + 1^2) = sqrt(2)
Substituting these values into the dot product formula gives:
(1)(1) + (1)(0) + (k)(1) = sqrt(2 + k^2) sqrt(2) cos(45)
Simplifying this equation, we have:
1 + k = sqrt(2 + k^2)
To solve for k, we need to square both sides of the equation:
(1 + k)^2 = (sqrt(2 + k^2))^2
Expanding and simplifying:
1 + 2k + k^2 = 2 + k^2
Rearranging the equation:
2k = 1
k = 1/2
Therefore, the value of k for which the angle between the vectors a = (1, 1, k) and b = (1, 0, 1) is 45 degrees is k = 1/2.
To determine the value(s) of k such that the angle between the vectors a = (1, 1, k) and b = (1, 0, 1) is 45 degrees, we can start by finding the dot product of the two vectors.
The dot product of two vectors can be found using the formula:
a · b = |a| * |b| * cos(theta)
where a · b is the dot product, |a| and |b| are the magnitudes (lengths) of the vectors a and b, and theta is the angle between the vectors.
Let's calculate the dot product of vectors a and b:
a · b = (1 * 1) + (1 * 0) + (k * 1)
= 1 + 0 + k
= k + 1
Now, let's find the magnitudes of vectors a and b:
|a| = sqrt(1^2 + 1^2 + k^2)
|b| = sqrt(1^2 + 0^2 + 1^2)
= sqrt(2)
Since the angle between a and b is given as 45 degrees, the cosine of 45 degrees is sqrt(2)/2. Therefore, we can rewrite the dot product formula as:
k + 1 = |a| * |b| * sqrt(2)/2
k + 1 = sqrt(2) * sqrt(2)/2
k + 1 = sqrt(2)/sqrt(2)
k + 1 = 1
To solve for k:
k = 1 - 1
k = 0
Therefore, the value of k that satisfies the condition of the angle between vectors a and b being 45 degrees is k = 0.