Find the sum of the given vectors and illustrate geometrically. [-2,-1],[5,7]
1 answer:
Answer:
The sum of vectors is <3,6>.
Step-by-step explanation:
The given vectors are <-2,-1> and <5,7>.
We need to find the sum of the given vectors and illustrate geometrically.
Plot the point (-2,-1) on a coordinate plane and draw a vector <a> from (0,0) to (-2,-1).
Plot the point (5,7) on a coordinate plane and draw a vector <b> from (0,0) to (5,7).
Now complete the parallelogram and the diagonal represents the sum of both vectors.
The end point of the diagonal is (3,6). It means sum of vectors is <3,6>.
Check the sum of vectors:
Therefore, the sum of vectors is <3,6>.
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Answer:
- x = 8√3
- y = z = 12√2
Step-by-step explanation:
We presume you want the values of x, y, and z.
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There are two "special triangles" in geometry and trigonometry. They are the 30°-60°-90° right triangle that is half of an equilateral triangle, and the 45°-45°-90° isosceles right triangle that is half a square (cut by the diagonal).
The side ratios of these special triangles are relatively easy to remember. It is useful to memorize them.
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For the isosceles right triangle, the side lengths are the same. The Pythagorean theorem tells you that if they are both 1, then the hypotenuse is ...
√(1²+1²) = √2
That is, the side lengths of the 45-45-90 triangle are in the ratio ...
1 : 1 : √2
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For the triangle that is half an equilateral triangle, you know the hypotenuse is twice the length of the shortest side (since we got that short side by cutting a long side in half). Then the longer side can be found from the Pythagorean theorem:
√(2²-1²) = √3
That is, the side lengths of the 30-60-90 triangle are in the ratio ...
1 : √3 : 2
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In this problem, we're given the hypotenuse of a 30-60-90 triangle, so we know the short side of it (x) will be half that length:
x = (16√3)/2
x = 8√3
The hypotenuse of the 45-45-90 triangle will be √3 times x, so will be ...
long side of small triangle = (√3)(8√3) = 24
The shorter sides of that 45-45-90 triangle will be this value divided by the square root of 2, so are ...
y = z = 24/√2
We can multiply this by (√2)/(√2) to "rationalize the denominator".
y = z = 12√2
