Final answer:
To determine which proportion statements are true, we need to find the dot product of the given vectors. The correct answer is option 1. I only.
Step-by-step explanation:
To determine which proportion statements are true, we need to find the dot product of the given vectors.
The dot product of two vectors a and b is given by the formula a · b = a1b1 + a2b2 + a3b3. We can calculate the dot product of each pair of vectors:
a · b = (2)(-3) + (1)(0) + (-4)(1) = -6 + 0 - 4 = -10
a · c = (2)(-1) + (1)(-1) + (-4)(2) = -2 - 1 - 8 = -11
b · c = (-3)(-1) + (0)(-1) + (1)(2) = 3 - 0 + 2 = 5
Comparing these results to the proportion statements:
I. a · b = -10, which means -10 = -10, so statement I is true.
II. a · c = -11 and b · c = 5, which means -11 ≠ 5, so statement II is false.
III. a · b = -10 and b · c = 5, which means -10 ≠ 5, so statement III is false.
Therefore, the correct answer is option 1. I only.
a=(2,1,−4),b=(−3,0,1),c=(−1,−1,2)
Which proportion statements are true? (take dot product of vectors)
1. I only.
2. I and III.
3. II and III.
4. II only.