Find the area of the parallelogram with vertices k(1, 1, 1), l(1, 3, 2), m(5, 8, 2), and n(5, 6, 1).
1 answer:
The area can be found as the magnitude of the cross product of the vectors representing two adjacent sides.
area = || kl × kn || = || (0, 2, 1) × (4, 5, 0) ||
= || (-5, 4, -8) || = √105
_____
For the purpose here, kl = l - k = (1, 3, 2) - (1, 1, 1) = (0, 2, 1)
and nk = n - k = (5, 6, 1) - (1, 1, 1) = (4, 5, 0)
The cross product can be calculated using a suitable calculator. By hand, it is the determinant
where i, j, k are unit vectors in the x, y, z directions.
area = || kl × kn || = || (0, 2, 1) × (4, 5, 0) ||
= || (-5, 4, -8) || = √105
_____
For the purpose here, kl = l - k = (1, 3, 2) - (1, 1, 1) = (0, 2, 1)
and nk = n - k = (5, 6, 1) - (1, 1, 1) = (4, 5, 0)
The cross product can be calculated using a suitable calculator. By hand, it is the determinant
where i, j, k are unit vectors in the x, y, z directions.
You might be interested in
Answer:
22-18-25,
19-24-30,
7-24-25
Step-by-step explanation:
In a triangle with side lengths ,
, and
, any two sides must have a sum larger than the third.
Using this rule, the possible side lengths out of these choices that could form a triangle are:
22-18-25,
19-24-30,
7-24-25
Answer:
Step-by-step explanation:
If you plot the directrix and the focus, you can see that the focus is to the left of the directrix. Since a parabola ALWAYS wraps itself around the focus, our parabola opens sideways, to the left to be specific. The formula for the parabola that opens to the left is
We will solve this for x at the end. The negative is out front because it opens to the left. If it opened to the right, it would be positive.
The vertex of a parabola is exactly halfway between the focus and the directrix, so our vertex coordinates h and k are (3, 6). P is defined as the distance between the vertex and the directrix, or the vertex and the focus. Since the vertex is directly between both the directrix and the focus, each distance is the same. P = 1. Filling in what we have now:
which simplifies to
Now we will solve it for x.
and
so