Math 2

Find the triple (or mix) product of vectors 𝑎⃗ = (1, −2, 4), 𝑏⃗⃗ = (5, −2, 1), 𝑐⃗ = (1, 2, 0).

• 44

• 12

• 42

• 14

Vertices 𝐴(5; 1; −4), 𝐵(1; 2; −1), 𝐶(3; 3; −4), 𝑆(2; 2; 2) of a pyramid are given. Find area of its side ABC

• √3

• 3√3

• 3√2

• 3

• (−3; 5; 11)

• (3; 5; -11)

• (3; -5; -11)

• (−3; 5; -11)

Find area of the parallelogram, defined by the vectors 𝑎⃗ = 6ⅈ⃗ + 3𝑗⃗ − 2𝑘⃗ and 𝑏⃗ = 3ⅈ⃗ − 2𝑗⃗ + 6𝑘⃗

• 36

• 7

• 49

• 6

Find |𝑎̅× 𝑏̅|, if |𝑎⃗| = 5 and |𝑏⃗| = 10, 𝑎⃗ ⋅ 𝑏⃗ = 30;

• 70

• 50

• 30

• 40

Find angle between diagonals of the parallelogram defined by the vectors 𝑎⃗ = 2ⅈ + 𝑗 and 𝑏⃗⃗ = −𝑗 + 2𝑘.

• 0

Scalar or dot product of 𝑎⃗ = (−1, −2, 3), 𝑏⃗ = (0, −1, 2) is

• 8

• -4

• -8

• 6

Find all values of a such that vector b is a linear combination of the vectors a1, a2, and a3, where a1 = (1, -1, 1), a2 = (1, 2, 2), a3 = (2, 1, a) and b = (1, 5, 1)

• 𝑎 ≠ 2

• 𝑎 ≠ 3

• 𝑎 ≠ 1

• 𝑎 ≠ -2

Express b as a linear combination of the vectors a1, a2, and a3, where a1 = (2, -3, 4, 1), a2 = (1, 6, -1, 2), a3 = (-1, -1, 2, 3) and b = (3, -17, 17, 7), if it is possible

• not solution

• not possible

• b is linear combination

• b is not linear combination

Establish linearly dependence of the vectors a = (1, 0, 1, 0), b = (1, 1, 0, 0) and c = (1, 0 , 0, 1)

• 𝑟𝑎𝑛𝑘(𝐴) = 2

• 𝑟𝑎𝑛𝑘(𝐴) = 3

• 𝑟𝑎𝑛𝑘(𝐴) = 1

• 𝑟𝑎𝑛𝑘(𝐴) = 4