12.4_ Cross Product-solutions

1 6 0 but then i p q p r

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: = −1 0 6 k. 6 Consequently, ∆P QR has In this case, |a × b|2 = 49 . area = 15 . Consequently, v=± 3 6 2 i− j− k 7 7 7 . keywords: vectors, cross product area, triangle, parallelogram stiurca (mas7745) – 12.4: Cross Product – meth – (91825) 005 10.0 points Find a vector v orthogonal to the plane through the points P (2, 0, 0), Q(0, 4, 0), R(0, 0, 5) . 3 3. volume = 25 4. volume = 21 5. volume = 22 correct 1. v = 5, 10, 8 Explanation: For the parallelopiped determined by vectors a, b, and c its 2. v = 20, 2, 8 volume = |a · (b × c)| . But 3. v = 4, 10, 8 31 22 a · ( b × c) = 5. v = 20, 10, 8 correct Explanation: Because the plane through P , Q, R con− − → − → tains the vectors P Q and P R, any vector v orthogonal to both of these vectors (such as their cross product) must therefore be orthogonal to the plane. =3 23 44 − −2 3 24 4. v = 20, 5, 8 4 2 3 2 4 −2 2 2 2 4 . Consequently, Here volume = 22 . − → P Q = −2, 4, 0 , − → P R = −2, 0, 5 . keywords: determinant, cross product, scalar triple product, parallelopiped, volume, Consequently, − − →− → v = P Q × P R = 20, 10, 8 is othogonal to the plane through P, Q and R. 006 10.0 points Compute the volume of the parallelopiped determined by the vectors a= 3, 1, −2 , b= 007 Which of the following statements are true for all vectors a, b = 0? A. a × b + b × a = 0, B. |a × b|2 + |a · b|2 = |a|2|b|2 , C. if a × b = 0, then a 2, 2, 3 , 1. A and B only and c = 2, 4, 4 . 10.0 points 2. C only 1. volume = 23 3. A only 2. volume = 24 4. none of them b. stiurca (mas7745) – 12.4: Cross Product – meth – (91825) 4 so if a = 0 and b = 0, then 5. all of them correct |a × b| = 0 =⇒ sin θ = 0 . 6. B only Thus θ = 0, π . In this case, a is parallel to b. 7. A and C only keywords: 8. B and C only 008 Explanation: 10.0 points Compute the volume of the parallelopiped with adjacent edges A. TRUE: if a = a1 , a2 , a3 , PQ, b = b1 , b2 , b3 , PR, PS determined by vertices then a2 b2 a×b = a3 b3 i− a1 a3 b1 b3 j+ a1 a2 b1 b2 P (1, −1, 1) , k, Q(4, −5, −1) , R(2, 3, 2) , S (2, 3, 3) . 1. volume = 15 while b2 b3 a2 b×a = a3 i− b1 b3 a1 a3 j+ b1 b2 a1 a2 k. 2. volume = 14 3. volume = 16 correct On the other hand, for a 2 × 2 determinant, a b c d = ad − cb = − c d a b 4. volume = 13 . Consequently, a × b = −b × a . 5. volume = 12 Explanation: The parallelopiped is determined by the vectors − − → a = P Q = 3, −4, −2 , − → b = PR = |a × b| = |a||b| sin θ , while 1, 4, 1 , − → c = PS = B. TRUE: if θ, 0 ≤ θ ≤ π , is the angle between a, and b, then 1, 4, 2 . Thus its volume is given in terms of a scalar triple product by a · b = |a||b| cos θ . Thus V = | a · ( b × c) | . But 2 |a × b| + |a · b| 2 = |a|2 |b|2(sin2 θ + cos2 θ ) = |a|2 |b|2 . 3 −4 =3 4 1 4 2 1 4 1 1 a · ( b × c) = C. TRUE: if θ, 0 ≤ θ ≤ π , is the angle between a, and b, then |a × b| = |a||b| sin θ , −2 4 2 +4 11 12 −2 14 14 . stiurca (mas7745) – 12.4: Cross Product – meth – (91825) Consequently, volume = 16 . keywords: determinant, cross product, vector product, scalar triple product, parallelopiped, volume, 5...
View Full Document

Ask a homework question - tutors are online