If Vector a = (2, 2k), Vector b = (4, -k), are perpendicular, solve for k
with solutions?
well, you need a•b = 0, so
(2)(4)+(2k)(-k) = 0
. . .
Well, it seems that Vector a and Vector b want to be best friends from a distance...perpendicular friends, to be exact. Let's see if we can help their friendship blossom.
To determine if two vectors are perpendicular, we need to check if their dot product equals zero. So, let's calculate the dot product of Vector a and Vector b:
(a•b) = (2)(4) + (2k)(-k) = 8 - 2k^2
Since we want Vector a and Vector b to be perpendicular, we need their dot product to equal zero. So, let's set it equal to zero and solve for k:
8 - 2k^2 = 0
Now, let's solve this equation to find the values of k that keep a smile on Vector a and Vector b's faces:
-2k^2 = -8
k^2 = 4
k = ±2
Ah, it seems that k has a couple of options to choose from. So, the solutions are k = 2 and k = -2. Now Vector a and Vector b can happily celebrate their perpendicular friendship!
To determine if two vectors are perpendicular, we can use the dot product. If the dot product of two vectors is zero, then they are perpendicular.
The dot product of two vectors a and b is given by:
a · b = (2)(4) + (2k)(-k) = 8 - 2k^2
Since the vectors are perpendicular, the dot product is equal to zero:
8 - 2k^2 = 0
Simplifying the equation:
2k^2 = 8
Dividing both sides of the equation by 2:
k^2 = 4
Taking the square root of both sides:
k = ±√4
Thus, the solutions for k are k = 2 and k = -2.
To determine if two vectors are perpendicular, we can use the dot product of the vectors. Two vectors are perpendicular if and only if their dot product is equal to zero.
Given: Vector a = (2, 2k) and Vector b = (4, -k).
The dot product between two vectors, denoted as "·", is calculated as follows:
Vector a · Vector b = (a1 * b1) + (a2 * b2)
Substituting the given values, we have:
(2 * 4) + (2k * -k) = 8 - 2k^2
For the vectors to be perpendicular, the dot product should be zero:
8 - 2k^2 = 0
Now we solve this equation for k:
2k^2 = 8
Divide both sides by 2:
k^2 = 4
Taking the square root of both sides (considering both positive and negative roots):
k = ± 2
Therefore, the values of k for which vector a and vector b are perpendicular are k = 2 and k = -2.