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
.

Question

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
.

BlackTom AI · Accepted Answer

We are asked to find a vector orthogonal to both a = ⟨−2, −5, 1⟩ and b = ⟨1, 3, −2⟩.

A standard way is to compute the cross product a × b, which yields a vector perpendicular to both a and b.

Compute a × b: 
Using the determinant form,
a × b = ⟨(−5)(−2) − (1)(3), −[(−2)(−2) − (1)(1)], (−2)(3) − (−5)(1)⟩
= ⟨10 − 3, −(4 − 1), −6 + 5⟩
= ⟨7, −3, −1⟩.

The vector ⟨7, −3, −1⟩ is orthogonal to both a and b by construction. Note that any nonzero scalar multiple of this vector is also orthogonal to both a and b, so ⟨−7, 3, 1⟩ (the negative of the cross product) is equally valid as a vector perpendicular to both.

If you compute the cross product in the opposite order, b × a, you would get the negative of a × b, namely ⟨−7, 3, 1⟩, which is still orthogonal to both original vectors. In summary, the key idea is that the cross product (in either order) provides a vector perpendicular to both given vectors, and the specific sign depends on the order of the cross product used.

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 .Single choice

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