MATH 21200 section 12.4 The Cross Product Page 1 The Cross Product of Two Vectors in Space Definition The cross product (“ cross u v × = × u v K K u = u K v = v K ”) is the vector ( sin ) or ( sin ) u v uv n θ θ × = × = u v uv n K K KK K where and are not parallel vectors and u = u K v = v K n = n K is the unit vector perpendicular to the plane (generated by and ) by the right-hand rule. u = u K v = v K Parallel Vectors Nonzero vectors and are parallel if and only if u = u K v = v K or 0 u v × = × u v 0 = K K Properties of the Cross Product If , , are any vectors and , are scalars, then u = u K v = v K w = w K r s 1. ( ) ( ) ( )( ) r s rs × = × u v u v ( ) ( ) ( )( ) ru sv rs u v × = × K K K K 2. ( ) × + = × + × u v w u v u w ( ) u v w u v u w × + = × + × K K K K K K K 3. ( × =− × v u u v) ( ) v u u v × =− × K K K K 4. ( ) + × = × + × v w u v u w u ( ) v w u v u w u + × = × + × K K K K K K K 5. 0 0 × = 0 u 0 u × = K K K 6. ( ) ( ) ( ) × × = − u v w uwv uvw i i ( ) ( ) ( ) u v w uwv uvw × × = − K K K KKK KKK i i u v × = × u v K K Is the Area of a Parallelogram Because is a unit vector, the magnitude of n = n K u v × = × u v K K is sin sin or sin sin u v uv n uv θ θ θ × = = × = = u v uv n uv θ K K KK K KK Determinant Formula for u v × = × u v K K Calculating the Cross Product as a Determinant If 1 2 3 u u u = + + u i j k and 1 2 3 v v v = + + v i j k , then 2 3 1 3 1 2 1 2 3 2 3 1 3 1 2 1 2 3 u u u u u u u u u v v v v v v v v v × = = − + i j k u v i j k If and , then 1 2 3 u ui uj uk = + + K K K K 1 2 3 v vi vj vk = + + K K K K 2 3 1 3 1 2 1 2 3 2 3 1 3 1 2 1 2 3 i j k u u u u u u u v u u u i j k v v v v v v v v v × = = − + K K K K K K K K

The Cross Product of Two Vectors in Space Definition KK u

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Page 1: The Cross Product of Two Vectors in Space Definition KK u

MATH 21200 section 12.4 The Cross Product Page 1

The Cross Product of Two Vectors in Space Definition The cross product (“ cross u v× = ×u v u=u v=v ”) is the vector ( sin ) or ( sin )u v u v nθ θ× = × =u v u v n where and are not parallel vectors and u=u v=v n=n is the unit vector perpendicular to the plane (generated by and ) by the right-hand rule. u=u v=v Parallel Vectors Nonzero vectors and are parallel if and only if u=u v=v or 0u v× = ×u v 0 = Properties of the Cross Product If , , are any vectors and , are scalars, then u=u v=v w=w r s 1. ( ) ( ) ( )( )r s rs× = ×u v u v ( ) ( ) ( )( )ru sv rs u v× = × 2. ( )× + = × + ×u v w u v u w ( )u v w u v u w× + = × + × 3. (× = − ×v u u v) ( )v u u v× = − × 4. ( )+ × = × + ×v w u v u w u ( )v w u v u w u+ × = × + × 5. 0 0× =0 u 0 u× = 6. ( ) ( ) ( )× × = −u v w u w v u v wi i ( ) ( ) ( )u v w u w v u v w× × = −i i

u v× = ×u v Is the Area of a Parallelogram Because is a unit vector, the magnitude of n=n u v× = ×u v is sin sin or sin sinu v u v n u vθ θ θ× = = × = =u v u v n u v θ Determinant Formula for u v× = ×u v Calculating the Cross Product as a Determinant

If 1 2 3u u u= + +u i j k and 1 2 3v v v= + +v i j k , then 2 3 1 3 1 21 2 3

2 3 1 3 1 21 2 3

u u u u u uu u u

v v v v v vv v v

× = = − +i j k

u v i j k

If and , then 1 2 3u u i u j u k= + + 1 2 3v v i v j v k= + + 2 3 1 3 1 21 2 3

2 3 1 3 1 21 2 3

i j ku u u u u u

u v u u u i j kv v v v v v

v v v× = = − +

Page 2: The Cross Product of Two Vectors in Space Definition KK u

MATH 21200 section 12.4 The Cross Product Page 2

Triple Scalar or Box Product The product ( ) is called triple scalar product (box product)of , ( )u v w× = ×u v wi i u=u v=v , and w=w (in that order). ( ) cos ( ) cosor u v w u v wθ θ× = × × = ×u v w u v wi i Calculating the Triple Scalar Product as a Determinant

1 2 3

1 2

( ) ( )u u u

u v w v v vw w w

× = × =u v wi i