For which value of
𝑐
will the vectors
𝑢
=
(
2

3
𝑐

−
4

1

5
)
and
𝑣
=
(
6

−
1

3

7

2
𝑐
)
be orthogonal?

Question

For which value of 
𝑐
 will the vectors 
𝑢
=
(
2

3
𝑐

−
4

1

5
)
 and 
𝑣
=
(
6

−
1

3

7

2
𝑐
)
 be orthogonal?

BlackTom AI · Accepted Answer

I will break down the reasoning by examining each answer choice and testing it in the orthogonality condition.

First, restating the problem in my own words: we are given two vectors u and v, dependent on c, and we need c such that their dot product u · v equals zero.

Option A: c = -1
To check orthogonality, compute the dot product with c = -1. If the result is zero, then this option is viable. We substitute c = -1 into u and v (conceptually) and calculate u · v = 2*6 + (3c)*(-1) + (-4)*3 + 1*7 + 5*(2c). Plugging c = -1 gives 2*6 + (3*(-1))*(-1) + (-4)*3 + 1*7 + 5*(2*(-1)) = 12 + 3 - 12 + 7 - 10 = 0. The sum is zero, so this choice yields orthogonality.

Option B: c = 1
Test c = 1 by evaluating the dot product: 12 + (3*1)*(-1) + (-4)*3 + 7 + 5*(2*1) = 12 - 3 - 12 + 7 + 10 = 14. Since the result is not zero, u and v are not orthogonal for this c.

Option C: c = 3
Test c = 3: 12 + (3*3)*(-1) + (-4)*3 + 7 + 5*(2*3) = 12 - 9 - 12 + 7 + 30 = 28. This is not zero, so this value does not make u and v orthogonal.

Option D: No such value of c exists.
From the computations above, we found a specific c that makes the dot product zero (c = -1). Therefore, the statement that no such c exists is incorrect because there is at least one value that works.

Summary of the reasoning across options:
- The orthogonality condition is u · v = 0. With the given u and v, this becomes 2*6 + (3c)*(-1) + (-4)*3 + 1*7 + 5*(2c) = 12 - 3c - 12 + 7 + 10c = 7 + 7c. Setting equal to zero gives 7 + 7c = 0, hence c = -1.
- Option A yields zero dot product upon substitution, confirming correctness.
- Options B and C fail because their substitutions lead to nonzero dot products, demonstrating they do not satisfy orthogonality.
- Option D is contradicted by the explicit solution c = -1.

For which value of 𝑐 will the vectors 𝑢 = ( 2 3 𝑐 − 4 1 5 ) and 𝑣 = ( 6 − 1 3 7 2 𝑐 ) be orthogonal?单项选择题

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