determine whether the vectors u and v are parallel, orthogonal, or neither.
u=(5,3) v=(-10/4, -3/2)
v=(-10/4, -3/2) = (-5/2, -3/2)
= (-1/2)(5,3)
so u is a scalar multiple of v, and therefore they are parallel
To determine if the vectors u=(5,3) and v=(-10/4, -3/2) are parallel, orthogonal, or neither, we can calculate their dot product.
1. Calculate the dot product:
The dot product of two vectors can be defined as the sum of the products of their corresponding components.
u · v = (5)(-10/4) + (3)(-3/2)
= -50/4 - 9/2
= (-50 - 18)/4
= -68/4
= -17/1
2. Analyze the dot product:
If the dot product of two vectors is equal to zero, then the vectors are orthogonal. If the dot product is non-zero, the vectors are not orthogonal.
In this case, the dot product of u and v is not zero since -17/1 is not equal to zero.
3. Determine if the vectors are parallel or neither:
If the vectors are not orthogonal, we need to check if they are parallel. Two vectors are parallel if one is a scalar multiple of the other.
To determine if u and v are parallel, we can compare their components.
The x-component of u is 5, and the x-component of v is -10/4, which simplifies to -5/2.
The y-component of u is 3, and the y-component of v is -3/2.
u and v are parallel if the ratio of their corresponding components is constant. Let's check:
5 / (-5/2) = 5 × (-2/5) = -2
3 / (-3/2) = 3 × (-2/3) = -2
Since both ratios are equal to -2, we can conclude that u and v are parallel.
Therefore, the vectors u=(5,3) and v=(-10/4, -3/2) are parallel.
To determine whether two vectors u and v are parallel, orthogonal, or neither, we need to find the dot product of the two vectors.
1. Calculate the dot product of u and v:
dot_product = u · v = (5 * -10/4) + (3 * -3/2)
Calculating the dot product:
dot_product = (-50/4) + (-9/2)
dot_product = -25/2 - 9/2
dot_product = -34/2
dot_product = -17
2. Determine the relationship based on the dot product:
- If the dot product is zero (0), then the vectors u and v are orthogonal (perpendicular).
- If the dot product is non-zero:
- If the dot product is positive, then the vectors u and v are either parallel or not orthogonal.
- If the dot product is negative, then the vectors u and v are not parallel and not orthogonal.
In this case, the dot product of u and v is -17, which is negative. Therefore, the vectors u and v are not parallel and not orthogonal.
Please note that if the dot product were zero, the vectors would be orthogonal (perpendicular). And if the dot product were positive, the vectors would either be parallel or not orthogonal.