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Match the following (volumes/areas of parallelepipeds and triangles formed by linear combinations of vectors): (P) A parallelepiped with edges a, b, c has volume 2. Find the volume of the parallelepiped with edges 2(a x b), 3(b x c), (c x a). (Q) A parallelepiped with edges a, b, c has volume 5. Find the volume of the parallelepiped with edges 3(a+b), (b+c), 2(c+a). (R) A triangle with sides a, b has area 20. Find the area of the triangle with sides (2a+3b), (a-b). (S) A parallelogram with sides a, b has area 30. Find the area of the parallelogram with sides (a+b), a.

1. P -> 3; Q -> 4; R -> 1; S -> 1
2. P -> 3; Q -> 4; R -> 2; S -> 1
3. P -> 3; Q -> 4; R -> 2; S -> 2
4. P -> 3; Q -> 4; R -> 1; S -> 2

Correct answer: P -> 3; Q -> 4; R -> 2; S -> 1

Solution

P: [2(axb), 3(bxc), (cxa)] = 6[(axb),(bxc),(cxa)] = 6[a,b,c]² = 6*4 = 24 -> list item 3. Q: [3(a+b),(b+c),2(c+a)] = 6[(a+b),(b+c),(c+a)] = 6*2[a,b,c] = 12*5 = 60 -> list item 4. R: Area = (1/2)|(2a+3b)x(a-b)| = (1/2)|2(axa)-2(axb)+3(bxa)-3(bxb)| = (1/2)*5|axb| = 5*20 = 100 -> list item 2. S: |(a+b)xa| = |axa + bxa| = |bxa| = |axb| = 30 -> list item 1.

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