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.

Question

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.

BlackTom AI · Accepted Answer

The direction vector of the line of intersection is given by the cross product of the normals of the two planes, because the line is perpendicular to both normal vectors.

Relevant concepts:
- Normal of plane ax+by+cz=d is n=(a,b,c).
- The cross product n1 × n2 yields a vector perpendicular to both n1 and n2, i.e., parallel to the intersection line.
- Any scalar multiple of a direction vector is also a valid direction vector.

Step-by-step reasoning and calculations:
1. Identify normals:
   n1 = (4, 2, 4) from 4x+2y+4z=1,
   n2 = (2, -3, 3) from 2x-3y+3z=2.
2. Compute cross product n1 × n2:
   n1 × n2 = |i   j   k|
             |4   2   4|
             |2  -3   3|
   = i*(2*3 - 4*(-3)) - j*(4*3 - 4*2) + k*(4*(-3) - 2*2)
   = i*(6 + 12) - j*(12 - 8) + k*(-12 - 4)
   = (18, -4, -16).
   So a direction vector is v = (18, -4, -16).
3. We are given a direction vector of the form w = (1, w2, w3). Scale v so its first component is 1 by dividing by 18:
   w = (18/18, -4/18, -16/18) = (1, -4/18, -16/18).
   Simplify the second component: -4/18 = -2/9 ≈ -0.222...
4. Rounding to two decimal places: w2 = -0.22.

Why this is correct:
- The cross product produced a vector parallel to the intersection line; scaling to make the first component 1 gives the required w2. The exact fraction -2/9 rounds to -0.22.

Why other values would be incorrect:
- Any number not equal to -2/9 (or its equivalent decimal -0.222...) arises from arithmetic mistakes (wrong cross product, wrong scaling) or rounding improperly. Reversing the cross product sign yields the same w2 after scaling (sign-consistent), so -0.22 is unique to the correctly computed scaled direction.

Final confirmation: w2 = -0.22, which matches the provided answer.

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.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 → ?

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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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