Suppose the condition stated in this problem holds for the two vectors a and b. If the
sum a+b is perpendicular to the difference a-b then the dot product of these two vectors
is zero:
(a + b) · (a - b) = 0
Use the distributive property of the dot product to expand the left side of this equation. We
get:
a · a - a · b + b · a - b · b
But the dot product of a vector with itself gives the magnitude squared:
a · a = a2
x + a2
y + a2
z = a2
(likewise b · b = b2) and the dot product is commutative: a · b = b · a. Using these facts,
we then have
a2 - a · b + a · b + b2 = 0 ,
which gives:
a2 - b2 = 0 =) a2 = b2
Since the magnitude of a vector must be a positive number, this implies a = b and so vectors
a and b have the same magnitude.
If the lines have the same slope but with different y intercepts then they are parallel
The circumcircle of a triangle is the circle that passes through the three vertices. Its center is at the circumcenter, which is the point O, at which the perpendicular bisectors of the sides of the triangle are concurrent. Since our triangle ABC is an isosceles triangle, the perpendicular line to the base BC of the triangle passes through the vertex A, so that OA (the part of the bisector perpendicular line to BC) is a radius of the circle O. Since the tangent line at A is perpendicular to the radius OA, and the extension of OA is perpendicular to BC, then the given tangent line must be parallel to BC (because two or more lines are parallel if they are perpendicular to the same line).
A Hypothesis is something that you set out to test to prove or disprove. A Lemma is something that has already been proved that you use to help prove something else.
Railway lines are parallel. 2 lines are said to be parallel when they are contained in the same plane and do not intersect. This is the definition. That parallel lines exist is an assumption (postulate) of Euclidean geometry:Parallel lines are like the rails of a train track, and you might think of defining them this way, as lines that are the same distance apart everywhere. The problem with this kind of definition is it assumes both tracks are straight. Though this seems an obvious possibility, when you go into the vast universe it is not that obvious. Parallel lines puzzled the best mathematicians for centuries until it was realized that we must assume they exist (you can't prove they exist from simpler postulates). The problem with parallel lines lies in the notion that the lines have infinite extent.Euclid used a somewhat different parallel postulate in trying to avoid the notion of the infinite. He observed that when two parallel lines are intersected by a third line, called a transversal, then if you measure two angles formed by these three lines, on the same side of the transversal and between the parallels, they will add to (that is, they will be supplementary). Such angles are called same-side interior angles.Another important concept is perpendicular. By definition, two lines are perpendicular if they intersect at right angles. That is, two perpendicular lines form 4 right angles. Segments and rays can also be perpendicular. This means they intersect in at least one point, and the two lines containing them are perpendicular.We use perpendicular segments to measure the distance from a point to a line, a point to a plane, or the distance between two parallel lines or planes. The ties of a railroad track are perpendicular to the rails and of the same length. This common length is the distance between the rails. (If parallel lines exist, then railroad tracks in space can go on forever.)There are three theorems about perpendicular lines that you should know. We will not attempt to prove them here, but if you think about them they should be rather obvious.We can use this fact to define the distance from a point to a line: That distance is the length of a segment perpendicular to the line with the given point as one of its endpoints and the other endpoint on the line. In fact, a similar notion holds in 3 dimensions. If we have a plane and a point not on that plane, then there is only one line through the point perpendicular to the plane, and the length of the segment determined by that point and the intersection of the perpendicular line with the plane is defined as the distance from the point to.
This is impossible to prove, as the square root of 2 is irrational.
That is not even true!
If the sum of their components in any two orthogonal directions is zero, the resultant is zero. Alternatively, show that the resultant of any two vectors has the same magnitude but opposite direction to the third.
You could draw a circle [center at origin] with radius of (a + b), for the two magnitudes a and b. This represents the sum of the magnitudes. Then draw one of the vectors starting at the origin [suppose it's vector a], and then draw a circle centered at the endpoint of vector a, with a radius of b. Drawing a circle demonstrates how the second vector can point in any direction relative to the first vector. The distance from the origin to a point on this second circle is the magnitude of the resultant vector. Graphically this second circle will be entirely inside the first circle and touching it at just one point. Since it lies within the first circle, the distance from the origin to a point on that circle will be less than or equal to the radius of the first circle.
Draw a perpendicular to that line and extend the arms of the angle to meed the perpendicular drawn earlier. Check if the line is bisecting the perpendicular, if yes, then the line is a bisector of the angle. :)
there is no difference
I will outline a way to prove it for you. I will also five a simple vector proof for those that have studied vectors. For the first proof, one can often cite some of these as known facts or refer to theorems in a text. 1. First show that a rhombus is a parallelogram 2. Next, using the above, show that diagonals of the rhombus divide it into 4 congruent triangles. 3. Last, use CPCTC and not that all 4 middle angles are congruent so that are 90 degrees. From this is is easy to say that the diagonals are perpendicular. Hints. to prove 1, use the fact that all 4 sides of the rhombus are congruent and then use SSS to find two congruent triangles. Then use CPCTC to show that the angles are the same and find a transversal. Look at same side interior angles cut by that transversal and say something about them being parallel. 2. Use SSS again and find 4 congruent triangles and look at the diagonals. I will help more by giving you another proof using vectors that is really much more straightforward. A rhombus is a quadrilateral with all sides having equal length. This means that if two vectors, a and b that form the corner of a rhombus, then the magnitude of a and b are equal The diagonals of the parallelogram are precisely a+b and a-b. Now look at the dot product of a+b and a-b and see that it is zero and remember that a dot product of zero means the vectors are perpendicular or orthogonal The first part is a pure synthetic geometry approach and if anyone need more help to finish that, just ask, The second part is a vector proof which is elegant because it is so simple.
If the vectors emanting from one corner of the rectangel are called a and b then. (a) + (b) = one diagonal (a) + (-b) = the other diagonal and |(a) + (b)| = |(a) + (-b)| (the absolute value of the diagonal's scalars are equal)
Show that corresponding angles are congruent?
The question is not correct, because the product of any two vectors is just a number, while when you subtract to vectors the result is also a vector. So you can't compare two different things...
A: By applying two known DC input with a known gain the output will be there to prove the difference.
In analytical geometry (geometry with numbers for coordinates), the easiest method is to show that they have the same slope.You could also prove that the distance between the lines, at different parts, is the same (draw a perpendicular to one of the lines).In analytical geometry (geometry with numbers for coordinates), the easiest method is to show that they have the same slope.You could also prove that the distance between the lines, at different parts, is the same (draw a perpendicular to one of the lines).In analytical geometry (geometry with numbers for coordinates), the easiest method is to show that they have the same slope.You could also prove that the distance between the lines, at different parts, is the same (draw a perpendicular to one of the lines).In analytical geometry (geometry with numbers for coordinates), the easiest method is to show that they have the same slope.You could also prove that the distance between the lines, at different parts, is the same (draw a perpendicular to one of the lines).
Head lice are currently being considered as vectors (able to carry disease). At the moment though there is nothing to prove that lice can carry or transmit virus or disease to humans..