Let
𝐴
=
(
1
,
5
,
−
2
)
,
𝐵
=
(
1
,
1
,
2
)
,
𝐶
=
(
3
,
2
,
5
)
be points in
𝑅
3
.

Which of the following vectors is orthogonal to both
𝐴
𝐵
→
and
𝐴
𝐶
→
?

Question

Let
𝐴
=
(
1
,
5
,
−
2
)
,
𝐵
=
(
1
,
1
,
2
)
,
𝐶
=
(
3
,
2
,
5
)
be points in 
𝑅
3
.

Which of the following vectors is orthogonal to both 
𝐴
𝐵
→
 and 
𝐴
𝐶
→
?

BlackTom AI · Accepted Answer

The cross product of AB and AC gives a vector orthogonal to both AB and AC, so compute AB, AC, then AB × AC.

Relevant concepts:
- Vector difference: given points A, B, C in R^3, AB = B − A and AC = C − A.
- Cross product: for u = (u1,u2,u3) and v = (v1,v2,v3), u × v = (u2 v3 − u3 v2, u3 v1 − u1 v3, u1 v2 − u2 v1). The result is perpendicular to both u and v.
- Orthogonality check: dot product zero.

Step-by-step computation:
1. Compute AB and AC.
   A = (1, 5, −2)
   B = (1, 1, 2)
   C = (3, 2, 5)

AB = B − A = (1−1, 1−5, 2−(−2)) = (0, −4, 4)
   AC = C − A = (3−1, 2−5, 5−(−2)) = (2, −3, 7)

2. Compute AB × AC using the determinant formula:
   AB × AC = | i   j   k |
             | 0  −4   4 |
             | 2  −3   7 |

Component-by-component:
   i-component = (−4)*7 − 4*(−3) = −28 + 12 = −16
   j-component = −[0*7 − 4*2] = −[0 − 8] = −(−8) = 8
   k-component = 0*(−3) − (−4)*2 = 0 + 8 = 8

So AB × AC = (−16, 8, 8).

3. Verify orthogonality by dot products:
   AB · (−16,8,8) = 0*(−16) + (−4)*8 + 4*8 = 0 − 32 + 32 = 0
   AC · (−16,8,8) = 2*(−16) + (−3)*8 + 7*8 = −32 − 24 + 56 = 0

Why this is correct:
- The cross product construction guarantees perpendicularity; the computed vector (−16, 8, 8) yields zero dot products with both AB and AC, confirming orthogonality.

Why other choices would be incorrect:
- Any correct choice must be a nonzero scalar multiple of (−16, 8, 8). Vectors not scalar multiples of this result will not be orthogonal to both AB and AC.

Conclusion:
The vector orthogonal to both AB and AC is [−16, 8, 8], matching the provided correct answer.

Let 𝐴 = ( 1 , 5 , − 2 ) , 𝐵 = ( 1 , 1 , 2 ) , 𝐶 = ( 3 , 2 , 5 ) be points in 𝑅 3 . Which of the following vectors is orthogonal to both 𝐴 𝐵 → and 𝐴 𝐶 → ?单项选择题

类似问题

Let 𝐴 = ( 5 , − 4 , 1 ) , 𝐵 = ( 2 , − 1 , 5 ) , 𝐶 = ( − 4 , − 1 , − 5 ) be points in 𝑅 3 . Which of the following vectors is orthogonal to both 𝐴 𝐵 → and 𝐴 𝐶 → ?

Consider the two planes 4x+2y+4z=1 and 2x+-3y+3z=2 If a direction vector in the direction of the line of intersection has the components: w = (1,w2, w3 ) Give the value of w2 to 2 decimal places, rounding if needed.

What is the area of the triangle which has two sides defined by the vectors u=(2,2,1) v=(5,2,1) Enter your answer to two decimal places, rounding as appropriate.

For two vectors u=[1,-9,-5] and v=[6,1,1] If the vector w=[w1,w2,w3] is defined as w=uxv give the value of w3.

Let 𝐴 = ( 2 , − 2 , 2 ) , 𝐵 = ( − 5 , − 4 , 5 ) , 𝐶 = ( − 4 , 0 , − 2 ) be points in 𝑅 3 . Which of the following vectors is orthogonal to both 𝐴 𝐵 → and 𝐴 𝐶 → ?

A vector that is orthogonal (perpendicular) to both vectors 𝑎 _ = ⟨ − 2 , − 5 , 1 ⟩ and 𝑏 _ = ⟨ 1 , 3 , − 2 ⟩ is: Hint: Vectors 𝑣 _ and 𝑤 _ are orthogonal if and only if 𝑣 _ ⋅ 𝑤 _ = 0 .

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