Let u = < − 2 , 1 , − 1 > , v = < − 3 , 2 , − 1 > , w = < 1 , 3 , 5 > u=<-2,1,-1>, v=<-3,2,-1>, w=<1,3,5> u =< − 2 , 1 , − 1 > , v =< − 3 , 2 , − 1 > , w =< 1 , 3 , 5 > .
Compute:
a) u × w = ∣ i j k − 2 1 − 1 1 3 5 ∣ = i ( 1 ⋅ 5 − ( − 1 ) ⋅ 3 ) − j ( ( − 2 ) ⋅ 5 − ( − 1 ) ⋅ 1 ) + k ( ( − 2 ) ⋅ 3 − 1 ⋅ 1 ) = i ( 5 + 3 ) − j ( − 10 + 1 ) + k ( − 6 − 1 ) = < 8 ; 9 ; − 7 > u\times w=
\begin{vmatrix}
i &j &k \\
-2&1&-1\\
1&3&5
\end{vmatrix}=i (1·5 - (-1)·3) - j ((-2)·5 - (-1)·1) + k ((-2)·3 - 1·1)
= i (5 + 3) - j (-10 + 1) + k (-6 - 1) = <8; 9; -7> u × w = ∣ ∣ i − 2 1 j 1 3 k − 1 5 ∣ ∣ = i ( 1 ⋅ 5 − ( − 1 ) ⋅ 3 ) − j (( − 2 ) ⋅ 5 − ( − 1 ) ⋅ 1 ) + k (( − 2 ) ⋅ 3 − 1 ⋅ 1 ) = i ( 5 + 3 ) − j ( − 10 + 1 ) + k ( − 6 − 1 ) =< 8 ; 9 ; − 7 >
b)
u × ( w × v ) = ∣ i j k 1 3 5 − 3 2 − 1 ∣ = i ( 3 ⋅ ( − 1 ) − 5 ⋅ 2 ) − j ( 1 ⋅ ( − 1 ) − 5 ⋅ ( − 3 ) ) + k ( 1 ⋅ 2 − 3 ⋅ ( − 3 ) ) = i ( − 3 − 10 ) − j ( − 1 + 15 ) + k ( 2 + 9 ) = u\times (w\times v)=\begin{vmatrix}
i&j&k \\
1&3&5\\
-3&2&-1
\end{vmatrix}=i (3·(-1) - 5·2) - j (1·(-1) - 5·(-3)) + k (1·2 - 3·(-3))
= i (-3 - 10) - j (-1 + 15) + k (2 + 9) = u × ( w × v ) = ∣ ∣ i 1 − 3 j 3 2 k 5 − 1 ∣ ∣ = i ( 3 ⋅ ( − 1 ) − 5 ⋅ 2 ) − j ( 1 ⋅ ( − 1 ) − 5 ⋅ ( − 3 )) + k ( 1 ⋅ 2 − 3 ⋅ ( − 3 )) = i ( − 3 − 10 ) − j ( − 1 + 15 ) + k ( 2 + 9 ) =
= ∣ i j k − 2 1 − 1 − 13 − 14 11 ∣ = i ( 1 ⋅ 11 − ( − 1 ) ⋅ ( − 14 ) ) − j ( ( − 2 ) ⋅ 11 − ( − 1 ) ⋅ ( − 13 ) ) + k ( ( − 2 ) ⋅ ( − 14 ) − 1 ⋅ ( − 13 ) ) = i ( 11 − 14 ) − j ( − 22 − 13 ) + k ( 28 + 13 ) = < − 3 ; 35 ; 41 > =\begin{vmatrix}
i&j&k \\
-2&1&-1\\
-13&-14&11
\end{vmatrix}= i (1·11 - (-1)·(-14)) - j ((-2)·11 - (-1)·(-13)) + k ((-2)·(-14) - 1·(-13))
= i (11 - 14) - j (-22 - 13) + k (28 + 13)= <-3; 35; 41> = ∣ ∣ i − 2 − 13 j 1 − 14 k − 1 11 ∣ ∣ = i ( 1 ⋅ 11 − ( − 1 ) ⋅ ( − 14 )) − j (( − 2 ) ⋅ 11 − ( − 1 ) ⋅ ( − 13 )) + k (( − 2 ) ⋅ ( − 14 ) − 1 ⋅ ( − 13 )) = i ( 11 − 14 ) − j ( − 22 − 13 ) + k ( 28 + 13 ) =< − 3 ; 35 ; 41 >
( u × w ) × v = ∣ i j k − 2 1 − 1 1 3 5 ∣ = i ( 1 ⋅ 5 − ( − 1 ) ⋅ 3 ) − j ( ( − 2 ) ⋅ 5 − ( − 1 ) ⋅ 1 ) + k ( ( − 2 ) ⋅ 3 − 1 ⋅ 1 ) = i ( 5 + 3 ) − j ( − 10 + 1 ) + k ( − 6 − 1 ) = ∣ i j k 8 9 − 7 − 3 2 − 1 ∣ = i ( 9 ⋅ ( − 1 ) − ( − 7 ) ⋅ 2 ) − j ( 8 ⋅ ( − 1 ) − ( − 7 ) ⋅ ( − 3 ) ) + k ( 8 ⋅ 2 − 9 ⋅ ( − 3 ) ) = i ( − 9 + 14 ) − j ( − 8 − 21 ) + k ( 16 + 27 ) = < 5 ; 29 ; 43 > . (u×w)×v=\begin{vmatrix}
i&j&k \\
-2&1&-1\\
1&3&5
\end{vmatrix}=i (1·5 - (-1)·3) - j ((-2)·5 - (-1)·1) + k ((-2)·3 - 1·1)
= i (5 + 3) - j (-10 + 1) + k (-6 - 1) =
\begin{vmatrix}
i&j&k\\
8&9&-7\\
-3&2&-1
\end{vmatrix}=i (9·(-1) - (-7)·2) - j (8·(-1) - (-7)·(-3)) + k (8·2 - 9·(-3))
= i (-9 + 14) - j (-8 - 21) + k (16 + 27) = <5; 29; 43>. ( u × w ) × v = ∣ ∣ i − 2 1 j 1 3 k − 1 5 ∣ ∣ = i ( 1 ⋅ 5 − ( − 1 ) ⋅ 3 ) − j (( − 2 ) ⋅ 5 − ( − 1 ) ⋅ 1 ) + k (( − 2 ) ⋅ 3 − 1 ⋅ 1 ) = i ( 5 + 3 ) − j ( − 10 + 1 ) + k ( − 6 − 1 ) = ∣ ∣ i 8 − 3 j 9 2 k − 7 − 1 ∣ ∣ = i ( 9 ⋅ ( − 1 ) − ( − 7 ) ⋅ 2 ) − j ( 8 ⋅ ( − 1 ) − ( − 7 ) ⋅ ( − 3 )) + k ( 8 ⋅ 2 − 9 ⋅ ( − 3 )) = i ( − 9 + 14 ) − j ( − 8 − 21 ) + k ( 16 + 27 ) =< 5 ; 29 ; 43 > .
Comments