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.

Question

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.

BlackTom AI · Accepted Answer

The cross product's third component is u1*v2 − u2*v1, so compute that directly.

1. Relevant knowledge points and concepts
- The cross product u × v for vectors u = [u1,u2,u3] and v = [v1,v2,v3] is given by
  u × v = [u2 v3 − u3 v2, u3 v1 − u1 v3, u1 v2 − u2 v1].
- The third component (w3) equals u1*v2 − u2*v1.
- Order matters: u × v ≠ v × u; reversing order negates the result.

2. Complete reasoning process step by step
- Identify components from the given vectors:
  u1 = 1, u2 = −9, u3 = −5;
  v1 = 6, v2 = 1, v3 = 1.
- Apply the formula for the third component:
  w3 = u1*v2 − u2*v1.

3. Calculations with intermediate steps
- Compute u1*v2 = 1 * 1 = 1.
- Compute u2*v1 = (−9) * 6 = −54.
- Now w3 = 1 − (−54) = 1 + 54 = 55.

4. Why the correct answer is correct
- The third component formula comes from the determinant expansion of the 3×3 matrix with unit vectors i, j, k in the first row and u, v components in the next two rows. Substituting the given numeric components into u1*v2 − u2*v1 yields 55, so w3 must be 55.

5. Common mistakes / why other answers would be incorrect
- A frequent error is swapping the order (computing u2*v1 − u1*v2) which would give −55. Another mistake is forgetting signs for negative components; here u2 = −9 must be multiplied correctly, producing −54. Any result other than 55 arises from such sign or order errors.

6. Conclusion and confirmation
- Therefore the third component w3 = 55, which matches the provided answer.

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.简答题

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.

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 → ?

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 .

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单