If [tex]$A =4 i -5 j +3 k , B =2 i -10 j -7 k$[/tex] and [tex]$C =5 i +7 j -4 k$[/tex], deduce the values of:
i. [tex]$( A \times B ) \cdot C$[/tex] and [tex]$A \times( B \times C )$[/tex]
ii. unit vectors perpendicular to [tex]$A$[/tex] and lying in the plane of [tex]$B$[/tex] and [tex]$C$[/tex].
Answer (2)
Calculate the cross product of vectors A and B: A × B = 65 i + 34 j − 30 k .
Compute the scalar triple product: ( A × B ) ⋅ C = 683 .
Calculate the vector triple product: A × ( B × C ) = − 239 i + 11 j + 337 k .
Determine the unit vectors perpendicular to A and in the plane of B and C: ± ( 0.578 i − 0.027 j − 0.815 k ) .
Explanation
Problem Analysis We are given three vectors:
A = 4 i − 5 j + 3 k B = 2 i − 10 j − 7 k C = 5 i + 7 j − 4 k
We need to find:
i. ( A × B ) ⋅ C and A × ( B × C ) ii. Unit vectors perpendicular to A and lying in the plane of B and C .
Calculate A x B First, let's find A × B . The cross product of two vectors A = a 1 i + a 2 j + a 3 k and B = b 1 i + b 2 j + b 3 k is given by:
A × B = i a 1 b 1 j a 2 b 2 k a 3 b 3 = ( a 2 b 3 − a 3 b 2 ) i − ( a 1 b 3 − a 3 b 1 ) j + ( a 1 b 2 − a 2 b 1 ) k
So, for A = 4 i − 5 j + 3 k and B = 2 i − 10 j − 7 k :
A × B = i 4 2 j − 5 − 10 k 3 − 7 = (( − 5 ) ( − 7 ) − ( 3 ) ( − 10 )) i − (( 4 ) ( − 7 ) − ( 3 ) ( 2 )) j + (( 4 ) ( − 10 ) − ( − 5 ) ( 2 )) k
A × B = ( 35 + 30 ) i − ( − 28 − 6 ) j + ( − 40 + 10 ) k = 65 i + 34 j − 30 k
Calculate (A x B) . C Now, let's find ( A × B ) ⋅ C . The dot product of two vectors X = x 1 i + x 2 j + x 3 k and Y = y 1 i + y 2 j + y 3 k is given by:
X ⋅ Y = x 1 y 1 + x 2 y 2 + x 3 y 3
So, for A × B = 65 i + 34 j − 30 k and C = 5 i + 7 j − 4 k :
( A × B ) ⋅ C = ( 65 ) ( 5 ) + ( 34 ) ( 7 ) + ( − 30 ) ( − 4 ) = 325 + 238 + 120 = 683
Calculate B x C Next, let's find B × C . For B = 2 i − 10 j − 7 k and C = 5 i + 7 j − 4 k :
B × C = i 2 5 j − 10 7 k − 7 − 4 = (( − 10 ) ( − 4 ) − ( − 7 ) ( 7 )) i − (( 2 ) ( − 4 ) − ( − 7 ) ( 5 )) j + (( 2 ) ( 7 ) − ( − 10 ) ( 5 )) k
B × C = ( 40 + 49 ) i − ( − 8 + 35 ) j + ( 14 + 50 ) k = 89 i − 27 j + 64 k
Calculate A x (B x C) Now, let's find A × ( B × C ) . For A = 4 i − 5 j + 3 k and B × C = 89 i − 27 j + 64 k :
A × ( B × C ) = i 4 89 j − 5 − 27 k 3 64 = (( − 5 ) ( 64 ) − ( 3 ) ( − 27 )) i − (( 4 ) ( 64 ) − ( 3 ) ( 89 )) j + (( 4 ) ( − 27 ) − ( − 5 ) ( 89 )) k
A × ( B × C ) = ( − 320 + 81 ) i − ( 256 − 267 ) j + ( − 108 + 445 ) k = − 239 i + 11 j + 337 k
Find vector V Now, let's find the unit vectors perpendicular to A and lying in the plane of B and C . Let V = m B + n C be a vector in the plane of B and C . Since V is perpendicular to A , A ⋅ V = 0 .
A ⋅ V = A ⋅ ( m B + n C ) = m ( A ⋅ B ) + n ( A ⋅ C ) = 0
First, we need to calculate A ⋅ B and A ⋅ C .
A ⋅ B = ( 4 ) ( 2 ) + ( − 5 ) ( − 10 ) + ( 3 ) ( − 7 ) = 8 + 50 − 21 = 37
A ⋅ C = ( 4 ) ( 5 ) + ( − 5 ) ( 7 ) + ( 3 ) ( − 4 ) = 20 − 35 − 12 = − 27
So, 37 m − 27 n = 0 . We can express m in terms of n :
37 m = 27 n ⟹ m = 37 27 n
Let's choose n = 37 , then m = 27 . Thus, V = 27 B + 37 C .
V = 27 ( 2 i − 10 j − 7 k ) + 37 ( 5 i + 7 j − 4 k ) = ( 54 i − 270 j − 189 k ) + ( 185 i + 259 j − 148 k ) = 239 i − 11 j − 337 k
Calculate unit vector Now, we need to find the magnitude of V :
∣ V ∣ = V ⋅ V = ( 239 ) 2 + ( − 11 ) 2 + ( − 337 ) 2 = 57121 + 121 + 113569 = 170811 ≈ 413.29
Finally, the unit vector V ^ is given by:
V ^ = ∣ V ∣ V = 170811 239 i − 11 j − 337 k = 170811 239 i − 170811 11 j − 170811 337 k
V ^ ≈ 0.578 i − 0.027 j − 0.815 k
The other unit vector is − V ^ ≈ − 0.578 i + 0.027 j + 0.815 k
Final Answer Therefore, we have:
i. ( A × B ) ⋅ C = 683 and A × ( B × C ) = − 239 i + 11 j + 337 k ii. The unit vectors perpendicular to A and lying in the plane of B and C are approximately 0.578 i − 0.027 j − 0.815 k and − 0.578 i + 0.027 j + 0.815 k .
Examples
Understanding vector operations like cross and dot products is crucial in various fields. For instance, in physics, the cross product helps determine the torque exerted by a force, while the dot product can calculate the work done by a force. In computer graphics, these operations are used to calculate surface normals for lighting and shading effects, creating realistic 3D models and scenes. Additionally, in robotics, vector operations are essential for controlling robot movements and ensuring precise interactions with the environment. These applications highlight the practical importance of mastering vector algebra.
We computed the scalar and vector triple products for the given vectors, finding ( A × B ) ⋅ C = 683 and A × ( B × C ) = − 239 i + 11 j + 337 k . Additionally, we determined the unit vectors perpendicular to A that lie in the plane of B and C as approximately 0.578 i − 0.027 j − 0.815 k and its negative.
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