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What is the gcf of 3ab3c and a3b2?
The GCF is ab2
Simplify the expression a3b2 over a2b?
1 over a^5b^3
What is the cross product of two vectors?
To find the cross product of two vectors:If a = [a1, a2, a3] and b = [b1, b2, b3], thena x b = [a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1] or(a2b3 - a3b2)i + (a3b1 - a1b3)j + (a1b2 - a2b1)k
What is the LCM of the numbers 24aaab and 36abb?
Prime factors of 24 are 23 x 3Prime factors of 36 are 22 x 32The LCM is the product of all the primes to their greatest power,so the LCM of 24aaab and 36abb is 23 x 32 x a3b2 = 72a3b2
Axb Dot bxc x cxa equals a Dot bxc squared How can I prove this using Mathmatica X stands for cross product and dot is or the dot product?
Prove that (axb)n[(bxc)x(cxa)] = [a]n(bxc)]^2 where a,b,and c are all vectors. First, multiply out the cross products. Since the cross product of two vectors is itself a vector, we'll give the cross products some names to make this a little easier to understand: (bxc)=(b2c3-b3c2)i-(b1c3-b3c1)j+(b1c2-b2c1)k = vector d (cxa)=(c2a3-c3a2)i-(c1a3-c3a1)j+(c1a2-c2a1)k = vector v (axb)=(a2b3-a3b2)i-(a1b3-a3b1)j+(a1b2-a2b1)k = vector u => (axb)n[(bxc)x(cxa)] = un[dxv] (dxv)=(d2v3-d3v2)i-(d1v3-d3v1)j+(d1v2-d2v1)k = vector w => un[dxv] = unw = u1w1 + u2w2 + u3w3 Now replace u and w with their vector coordinates (notice that the negative sign is factored into the middle terms, so the variables are switched). u1w1 + u2w2 + u3w3= (a2b3-a3b2)w1 + (a3b1-a1b3)w2 + (a1b2-a2b1)w3 = (a2b3-a3b2)(d2v3-d3v2) + (a3b1-a1b3)(d3v1-d1v3)+ (a1b2-a2b1)(d1v2-d2v1) Now we need to expand the v terms back out: (d2v3-d3v2) = d2(c1a2-c2a1) - d3(c3a1-c1a3) = d2c1a2- d2 c2a1- d3c3a1 + d3c1a3 (d3v1-d1v3) = d3(c2a3-c3a2) - d1(c1a2-c2a1) = d3c2a3 - d3c3a2- d1c1a2+ d1c2a1 (d1v2-d2v1) = d1(c3a1-c1a3) - d2(c2a3-c3a2) = d1c3a1 - d1c1a3 - d2c2a3 + d2c3a2 So: (a2b3-a3b2)(d2v3-d3v2) + (a3b1-a1b3)(d3v1-d1v3)+ (a1b2-a2b1)(d1v2-d2v1) = (a2b3-a3b2)(d2c1a2- d2 c2a1- d3c3a1 + d3c1a3) + (a3b1 - a1b3)(d3c2a3 - d3c3a2- d1c1a2+ d1c2a1)+ (a1b2-a2b1)(d1c3a1 - d1c1a3 - d2c2a3 + d2c3a2) = d2c1a2 a2b3- d2 c2a1 a2b3- d3c3a1 a2b3 + d3c1a3 a2b3- d2c1a2 a3b2+ d2 c2a1 a3b2+ d3c3a1 a3b2- d3c1a3 a3b2 + d3c2a3 a3b1 - d3c3a2 a3b1- d1c1a2 a3b1+ d1c2a1 a3b1- d3c2a3 a1b3+ d3c3a2 a1b3+ d1c1a2 a1b3- d1c2a1 a1b3+ d1c3a1 a1b2 - d1c1a3 a1b2 - d2c2a3 a1b2 + d2c3a2 a1b2- d1c3a1 a2b1+ d1c1a3 a2b1+ d2c2a3 a2b1- d2c3a2 a2b1 Some of the terms cancel out, leaving us with; = d2c1a2 a2b3 - d2 c2a1 a2b3 + d3c1a3 a2b3 - d2c1a2 a3b2 + d3c3a1 a3b2 - d3c1a3 a3b2 + d3c2a3 a3b1 - d3c3a2 a3b1 + d1c2a1 a3b1 - d3c2a3 a1b3 + d1c1a2 a1b3 - d1c2a1 a1b3 + d1c3a1 a1b2 - d1c1a3 a1b2 + d2c3a2 a1b2 - d1c3a1 a2b1 + d2c2a3 a2b1 - d2c3a2 a2b1 Now factor out d1 , d2 , and d3 = d1(c2a1 a3b1 + c1a2 a1b3 - c2a1 a1b3 + c3a1 a1b2 - c1a3 a1b2 - c3a1 a2b1) + d2(c1a2 a2b3 - c2a1 a2b3 - c1a2 a3b2 + c3a2 a1b2 + c2a3 a2b1 - c3a2 a2b1) + d3(c1a3 a2b3 + c3a1 a3b2 - c1a3 a3b2 + c2a3 a3b1 - c3a2 a3b1 - c2a3 a1b3) Now we can factor out a dot product of ( d1 + d2 + d3): = ( d1 + d2 + d3)n[(c2a1 a3b1 + c1a2 a1b3 - c2a1 a1b3 + c3a1 a1b2 - c1a3 a1b2 - c3a1 a2b1) + (c1a2 a2b3 - c2a1 a2b3 - c1a2 a3b2 + c3a2 a1b2 + c2a3 a2b1 - c3a2 a2b1) + (c1a3 a2b3 + c3a1 a3b2 - c1a3 a3b2 + c2a3 a3b1 - c3a2 a3b1 - c2a3 a1b3)] (Remember, to keep from changing the value of the equation we still need to keep the terms grouped together so that they multiply by the correct d components.) Now factor out all the "a" components within the brackets: = ( d1 + d2 + d3)n[(a1 a3{c2b1 - c1b2} + a1 a2{c1b3 - c3b1} + a1 a1{c3b2 - c2b3}) + (a1 a2{c3b2 - c2b3} + a2 a2{c1b3 - c3b1} + a2 a3{c2b1 - c1b2}) + (a1 a3{c3b2 - c2b3} + a2 a3{c1b3 - c3b1} + a3 a3{c2b1 - c1b2})] = dn[( a1 a3+ a1 a2+ a1 a1)n({c2b1- c1b2} +{c1b3 - c3b1} + {c3b2- c2b3}) + (a1 a2 + a2 a2 + a2 a3)n({c1b3- c3b1} + {c2b1- c1b2} + {c3b2- c2b3}) + ( a1 a3+ a2 a3 + a3 a3)n({c3b2- c2b3} + {c1b3- c3b1} + {c2b1- c1b2})] And we know that {c2b1- c1b2} +{c1b3 - c3b1} + {c3b2- c2b3} = (bxc), so we factor out (bxc): = dn[(bxc)n[(a1 a3+ a1 a2+ a1 a1) + (a1 a2 + a2 a2 + a2 a3) + ( a1 a3+ a2 a3 + a3 a3)] = dn[(bxc)n[a1(a3+ a2 + a1) + a2 (a1 + a2 +a3) + a3(a1+ a2 + a3)]] = dn[(bxc)n([a1 + a2 +a3]n[a1 + a2 +a3]) = dn[(bxc)n(a n a)] (from above, remember that d = (bxc) ) = (bxc)n(bxc)n a n a = [an (bxc)]^2
What is the LCM of a3b2 and a2b5?
The answer is a^2b^2, because the smallest exponent of the a's is 2 and the same thing with the b's. Therefore, that's the LCM (or least common multiple), because it is the smallest value the two terms share with one another. **When writing an exponent on a computer, you use a carrot (^) to represent the exponent.
How do you determine the degrees of a given polynomial?
The degree of a polynomial is the highest power that appears in the polynomial. For more than one variable, you must add the powers for each variable, for example, a3b2 is of degree 3 + 2 = 5.
How cross product is defined in n-dimensional vector space.this is asked by g. sudhakaran.his e-mai id is sudhakaranghotmail.com?
The cross product in n-dimensional vector space is the non-diagonal lements of the matrix created by the multiplicationof the two n-dimensional vectors. For example: A= a1 + a2 + a3 + a4 and B= b1 + b2 + b3 + b4 where A and B are vectors, then the cross product is simply the square multiplication table product minus the diagonal terms: AXB= b1 b2 b3 b4 ----------------------------- a1| a1b2 a1b3 a1b4 a2| a2b1 a2b3 a2b4 a3| a3b1 a3b2 a3b4 a4| a4b1 a4b2 a4b3 The cross product here is : (a1b2-a2b1) + (a1b3 - a3b1) + (a1b4 - a4b1) + (a2b3 -a3b2) + (a2b4-a4b2) + (a3b4-a4b3)
How do you do cross products?
You need to know that the cross product of two vectors is a vector perpendicular to both vectors. It is defined only in 3 space. The formula to find the cross product of vector a (vector a=[a1,a2,a3]) and vector b (vector b=[b1,b2,b3]) is: vector a x vector b = [a2b3-a3b2,a3b1-a1b3,a1b2-a2b1]
What is the chemical composition of garnet?
Garnet is a group of silicate minerals with the general formula X3Y2(SiO4)3, where X can be calcium, magnesium, ferrous iron, or manganese, and Y can be aluminum, ferric iron, or chromium. This composition gives different garnet species their unique colors and properties.
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.
