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The magnitude of dot product of two vectors is equal to the product of first vector to the component of second vector in the direction of first.
for ex.- A.B=ABcos@
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Prove that two vectors must have equal magnitude if their sum is perpendicular to their difference?
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.
What are the properties of a dot product?
In mathematics, the dot product is an algebraic operation that takes two equal-length sequences of numbers (usually vectors) and returns a single number obtained by multiplying corresponding entries and adding up those products. The name is derived from the interpunct "●" that is often used to designate this operation; the alternative name scalar product emphasizes the scalar result, rather than a vector result.The principal use of this product is the inner product in a Euclidean vector space: when two vectors are expressed in an Orthonormal basis, the dot product of their coordinate vectors gives their inner product. For this geometric interpretation, scalars must be taken to be Real. The dot product can be defined in a more general field, for instance the complex number field, but many properties would be different. In three dimensional space, the dot product contrasts with the cross product, which produces a vector as result.
What is dot product?
In vector calculus a dot product of two vectors is basically the product of the length of one vector and the length of the parallel component of the other; It doesn't matter which one is taken first because length is a scalar and scalars are commutative. the easiest way to determine the dot product of u and v(u•v) is to simply multiply the length of each vector together and then multiply by the cosine of the angle between them (|uv|cosӨ, because length is a scalar, the product is always a scalar). You could also identify the the component of v that is parallel to u and and multiply their lengths but it's basically the same thing (|v|cosӨ|u|).
How do you do rotation in math?
A vector rotation in math is done on a coordinate plane.2D vectors can be rotated using the cross and dot product.3D vectors are rotated using matrix based quaternion math.
What is the used of dot product and cross product in real life?
The dot-product and cross-product are used in high order physics and math when dealing with matrices or, for example, the properties of an electron (spin, orbit, etc.).
Why in dot product you use cos and in vector product sin?
We use the dot product cos and in vector we use the vector product sin because of the trigonometric triangle.
What is it when two vectors' dot product is one?
That fact alone doesn't tell you much about the original two vectors. It only says that (magnitude of vector-#1) times (magnitude of vector-#2) times (cosine of the angle between them) = 1. You still don't know the magnitude of either vector, or the angle between them.
What is cross-product and dot-product?
Cross products and dot products are two operations that can be done on a pair of 2-dimensional, 3-dimensional, or n-dimensional vectors. Both can be viewed in terms of mathematics or their physical representations.The dot product of two three-dimensional vectors A= and B= is a1b1+ a2b2 + a3b3. The definition in high dimensions is completely analogous. Notice that the dot product of two vectors is a scalar, not a vector. The dot product also equals |A|*|B|cosθ, where |A| and |B| are the magnitudes of A and B, respectively and θ is the angle between the vectors. This is the same as saying that the dot product is the magnitude of one vector multiplied times the component of the second vector that is parallel to the first. Notice that this means that the dot product of two vectors is 0 if and only if they are perpendicular.The cross product is a little more complicated. In three dimensions, A × B = . Notice that this operation results in another vector. This vector always points in a direction perpendicular to both A and B, and this direction can be determined by the right-hand rule. Physically, the magnitude of this vector equals |A|*|B|sinθ, or the magnitude of the first vector times the component of the other that is perpendicular to the first. So the cross product is 0 when the vectors are parallel.
Dot product of two vectors is equal to cross product what will be angle between them?
(A1) The dot product of two vectors is a scalar and the cross product is a vector? ================================== (A2) The cross product of two vectors, A and B, would be [a*b*sin(alpha)]C, where a = |A|; b = |B|; c = |C|; and C is vector that is orthogonal to A and B and oriented according to the right-hand rule (see the related link). The dot product of the two vectors, A and B, would be [a*b*cos(alpha)]. For [a*b*sin(alpha)]C to equal to [a*b*cos(alpha)], we have to have a trivial solution -- alpha = 0 and either a or b be zero, so that both expressions are zeroes but equal. ================================== Of course one is the number zero( scalar), and one is the zero vector. It is a small difference but worth mentioning. That is is to say if a or b is the zero vector, then a dot b must equal zero as a scalar. And similarly the cross product of any vector and the zero vector is the zero vector. (A3) The magnitude of the dot product is equal to the magnitude of the cross product when the angle between the vectors is 45 degrees.
What is vector dot product?
The dot-product of two vectors is the product of their magnitudes multiplied by the cosine of the angle between them. The dot-product is a scalar quantity.
in what components dot and cross product resolve?
A dot product is a scalar product so it is a single number with only one component. A cross product or vector product is a vector which has three components like the original vectors.
Where is the resultant vector lies when dot product of two vector is done?
arithmetic mean
Is triple dot product defined?
If by "triple dot product" you mean u·v·w, then no, because that would imply the existence of a dot product between a vector and a scalar.
What is the product of two vector quantities?
It depends on the type of product used. A dot or scalar product of two vectors will result in a scalar. A cross or vector product of two vectors will result in a vector.
Is work is the vector product of force and distance?
Yes and no. It's the dot product, but not the cross product.
What is the dot product of two rectangular components of a vector?
a vector is a line with direction and distance. there is no answer to your question. the dot is the angular relationship between two vectors.
Prove that two vectors must have equal magnitude if their sum is perpendicular to their difference?
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.
