Final answer:
To find a times b, use the dot product of the two vectors. To find two different unit vectors perpendicular to given vectors a and b, use the cross product. To find the area of a triangle, use the cross product to find the magnitude of the cross product of two sides. To find the area of a parallelogram, use the cross product to find the magnitude of the cross product of two adjacent sides.
Step-by-step explanation:
To find the value of a times b, you can use the dot product of the two vectors a and b. The dot product is calculated by multiplying the corresponding components of the vectors and adding them together. So, a · b = (2)(2) + (-4)(3) + (1)(1) = 4 - 12 + 1 = -7.
To find two different unit vectors u and v, you can use the cross product of the given vectors a = 3i + 12j and b = 4j + 3k. The cross product is calculated by finding a vector that is perpendicular to both a and b. u = a × b = (0)(3) - (3)(4) + (12)(1) = -12i - 12k. To find v, you can take the cross product of a and u. v = a × u = (3)(-12) - (12)(0) + (0)(-12) = -36j.
To find the area of the triangle with vertices P(1, 3, -2), Q(2, 4, 5), and R(3, -2, 2), you can use the formula for the area of a triangle in three-dimensional space. Area = 1/2 · |PR × PQ|, where |PR × PQ| represents the magnitude of the cross product of the vectors PR and PQ. PR = R - P = (3 - 1)i + (-2 - 3)j + (2 + 2)k = 2i - 5j + 4k and PQ = Q - P = (2 - 1)i + (4 - 3)j + (5 + 2)k = i + j + 7k. PR × PQ = (5)(7) - (4)(1) + (-5)(1) = 31. So, the area of the triangle is 1/2 · |31| = 15.5 square units.
To find the area of the parallelogram whose vertices are A(-6, 0), B(1, -4), C(3, 1), and D(-4, 5), you can use the formula for the area of a parallelogram in two-dimensional space. Area = |AB × AC|, where AB and AC are two adjacent sides of the parallelogram. AB = B - A = (1 - (-6))i + (-4 - 0)j = 7i - 4j and AC = C - A = (3 - (-6))i + (1 - 0)j = 9i + j. AB × AC = (7)(9) - (-4)(1) = 67. So, the area of the parallelogram is |67| = 67 square units.