Final answer:
To prove that vectors u and v are orthogonal, we calculate their dot product. If the dot product is zero, it indicates that the vectors are orthogonal. The calculation shows their dot product is zero, hence they are orthogonal.
Step-by-step explanation:
In order to prove that two vectors are orthogonal, we need to use the concept of the dot product. The dot product of two vectors u = ⟨2,3⟩ and v = ⟨3/2,-1⟩ gives a scalar value that measures the extent to which the two vectors point in the same direction. If the dot product of the two vectors is zero, this indicates that the vectors are orthogonal, meaning they are at right angles to each other in the plane.
To find the dot product of vectors u and v, we multiply the corresponding components and sum the products:
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u ⋅ v = (2) × (3/2) + (3) × (-1) = 3 - 3 = 0.
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As the dot product is zero, we can conclude that the vectors u and v are indeed orthogonal. This proves that there is a 90-degree angle between u and v.
Understanding that the vectors are orthogonal relates to the concept of a unit vector, which points in a specific direction and has a magnitude of one. However, the magnitude of a vector does not affect whether it is orthogonal to another vector. The key is the angle between them, which can be determined using their dot product.