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
→
?

Question

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
→
?

BlackTom AI · Accepted Answer

The cross product of AB and AC produces a vector perpendicular to both, so compute AB and AC and take their cross product.

Relevant concepts:
- Vector from A to B is AB = B − A; from A to C is AC = C − A.
- The cross product AB × AC yields a vector orthogonal to both AB and AC.
- Orthogonality can be checked by dot products: v is orthogonal to w if v · w = 0. Any nonzero scalar multiple of a cross product is also orthogonal to both original vectors.

Step-by-step computation:
1. Compute AB and AC:
   AB = B − A = (−5 − 2, −4 − (−2), 5 − 2) = (−7, −2, 3).
   AC = C − A = (−4 − 2, 0 − (−2), −2 − 2) = (−6, 2, −4).

2. Compute AB × AC using the determinant expansion:
   AB × AC = | i    j    k |
             | −7  −2   3 |
             | −6   2  −4 |

i·[ (−2)(−4) − (3)(2) ] − j·[ (−7)(−4) − (3)(−6) ] + k·[ (−7)(2) − (−2)(−6) ]
   = i·[ 8 − 6 ] − j·[ 28 + 18 ] + k·[ −14 − 12 ]
   = i·2 − j·46 + k·(−26)
   = (2, −46, −26).

3. Verify orthogonality by dot products:
   AB · (2,−46,−26) = (−7)(2) + (−2)(−46) + (3)(−26) = −14 + 92 − 78 = 0.
   AC · (2,−46,−26) = (−6)(2) + (2)(−46) + (−4)(−26) = −12 − 92 + 104 = 0.
   Both are zero, confirming orthogonality.

Why this answer is correct:
- The computed cross product (2, −46, −26) is orthogonal to both AB and AC by definition and verified by dot products.
- Any correct answer must be a nonzero scalar multiple of this vector; vectors not scalar multiples will not satisfy both dot products equaling zero, so they would be incorrect.

Conclusion: The vector orthogonal to both AB and AC is (2, −46, −26), which matches the provided answer.

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 → ?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 = ( 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 → ?

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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