To determine if two vectors are orthogonal, we need to check if their dot product is equal to zero.
The dot product (or scalar product) of two vectors u=(u1, u2) and v=(v1, v2) is calculated using the formula:
u · v = u1v1 + u2v2
In this case, the vectors u=(-6,4) and v=(3,3). Let's calculate their dot product:
u · v = (-6)(3) + (4)(3)
= -18 + 12
= -6
Since the dot product of u and v is not equal to zero (-6 ≠0), the vectors u and v are not orthogonal.
To find the angle between these two vectors, we can use the formula:
cosθ = (u · v) / (||u|| ||v||)
where ||u|| and ||v|| are the magnitudes (or lengths) of the vectors u and v, respectively.
The magnitudes can be calculated using the formula:
||u|| = √(u1^2 + u2^2)
||v|| = √(v1^2 + v2^2)
In this case:
||u|| = √((-6)^2 + 4^2)
= √(36 + 16)
= √52
= 2√13
||v|| = √(3^2 + 3^2)
= √(9 + 9)
= √18
= 3√2
Now we can substitute these values in the formula to find cosθ:
cosθ = (-6) / (2√13 * 3√2)
Simplifying the expression:
cosθ = -6 / (6√13√2)
= -1 / (√13√2)
= -1 / (√26)
To find the angle θ, we can take the inverse cosine (arccos) of cosθ. However, since -1 / (√26) is a decimal value, we can convert it to degrees or radians:
θ ≈ 137.6° or θ ≈ 2.4 radians
Therefore, the angle between the vectors u and v is approximately 137.6 degrees or 2.4 radians.