The dot product as a shadow

• basis vectors 1 and 2 are perpendicular ×12
• the polynomials of degree 0 and 1 are exactly perpendicular ×10
• the polynomial of degree 0 is monic ×5
• basis vector 1 has length one ×4
• the degree-0 result is the Legendre polynomial scaled to be monic ×4
• vector 1 is rebuilt from its own row of the table ×4
• moving coefficient 0 by -0.25 makes the total worse ×3
• moving coefficient 0 by 0.25 makes the total worse ×3
• the 1th vector uses no direction built after it ×3
• the residual is perpendicular to the column of x^0 ×3
• vector 1 is two finite coordinates no larger than twelve ×2
• a non-zero vector has positive weighted length ×1
• a plane and an arrow need the same number of coordinates only in three dimensions ×1
• a plane in seven dimensions needs twenty-one numbers ×1
• a rational is two whole numbers ×1
• a squared length is never negative ×1
• a, b and a × b, in that order, form a right-handed frame ×1
• a·b is |a||b| cos θ ×1
• a·b is the shadow's length times |b| ×1
• an ordinary fit is compared only against a weighted one ×1
• and holds in three ×1
• and in the second ×1
• and is not perpendicular in the ordinary sense unless the weights agree ×1
• and it is linear in its first slot ×1
• and its squared length is what Cauchy–Schwarz leaves over ×1
• and only there is the perpendicular to a plane a single line ×1
• and so has the second ×1
• and so it lies along the adjoint's kernel ×1
• and the weighted fit does worse under the ordinary one ×1
• and to the second ×1
• b is reachable exactly when it casts no shadow on the kernel ×1
• between one and three stages ×1
• between three and forty points ×1
• between two and five columns ×1
• between two and four vectors go in ×1
• both weights are positive and no larger than twenty-five ×1
• degree between one and five ×1
• each column is three finite coordinates ×1
• each of the three pairs multiplies to the fourth unit ×1
• every neighbouring point of the plane is further from the target ×1
• every pair of distinct units lies on exactly one line ×1
• every point is a finite pair ×1
• every point of the drawn curve has length one under the product ×1
• every point of the drawn curve has weighted length one ×1
• every vector has the same number of finite coordinates ×1
• every ε is between nought and one ×1
• everything the map reaches is perpendicular, under the product, to what the adjoint kills ×1
• its length is the parallelogram's area, measured in the parallelogram's own plane ×1
• its square and the square of the dot product add to |a|²|b|² ×1
• Lagrange's identity holds for every pair tried ×1
• moving the map across the product leaves the number where it was ×1
• multiplying the reflections gives minus the reflection of the product ×1
• no division by nought ×1
• no sampled value falls below the vertex ×1
• on the smallest ε drawn, the classical order loses orthogonality outright ×1
• removing a shadow never lengthens a vector ×1
• so the two arrows point into opposite half-spaces ×1
• some pair of vectors tells the transpose apart from the adjoint under this product ×1
• the adjoint of the adjoint is the map that started ×1
• the adjoint sends the drawn direction to nought ×1
• the camera angles are within a half turn ×1
• the classical order is never the better of the two ×1
• the cofactor matrix carries the old normal to the new one exactly ×1
• the count runs to between 4 and 10 dimensions ×1
• the degree is 1, 2 or 3 ×1
• the drawn direction is an eigenvector, in the first coordinate ×1
• the dropped line meets b at a right angle ×1
• the eigen-directions are named by two short strings, or not at all ×1
• the eigenvectors are perpendicular under the product exactly when the map is its own adjoint under it ×1
• the first basis vector has been scaled to length one ×1
• the first edge is three finite coordinates no larger than six ×1
• the first vector is not zero ×1
• the first vector is seven whole numbers under ten ×1
• the first vector is three finite coordinates no larger than six ×1
• the first vector is two finite coordinates no larger than twelve ×1
• the identity holds for every pair tried, not only for the drawn one ×1
• the inequality holds for the weighted product too ×1
• the inner product is positive definite, so every non-zero vector has a positive length ×1
• the inner product is three finite numbers [p, q, r] no larger than twenty-five ×1
• the Jacobi identity fails in seven dimensions ×1
• the least value is zero exactly when the two vectors are parallel ×1
• the map applied to the old normal is visibly not the new normal ×1
• the map has rank one, so what it reaches is a line ×1
• the map has two real eigenvalues ×1
• the map is four finite entries no larger than twelve ×1
• the map is invertible, so the patch stays a patch ×1
• the map is three rows of three finite entries no larger than four ×1
• the matrix being inverted is invertible ×1
• the mirror is a coordinate plane, named by the axis it reverses ×1
• the mode of the projection family is one of shadow, signs, schwarz, gram, qr, least, columns, weight, orthloss, legendre, adjoint, selfadjoint, ranges, cross, shadows, mirror, normal, planes, seven ×1
• the modified order stays near the arithmetic's own precision ×1
• the new normal is perpendicular to the first new edge ×1
• the normal equations have a solution ×1
• the nudged fit has one coefficient per column ×1
• the nudged line is worse than the fitted one ×1
• the ordinary fit does worse under the weighted product ×1
• the pair still spans the same area, so it spans the same plane ×1
• the parabola's least value is |a|² − (a·b)²/|b|² ×1
• the plotted span is between 0.2 and 6 ×1
• the product is perpendicular to the first factor ×1
• the product is perpendicular to the first vector ×1
• the quadratic cannot have two distinct real roots ×1
• the remainder is perpendicular to the line the map reaches ×1
• the residual is perpendicular to the first column ×1
• the second drawn vector is perpendicular in the weighted sense ×1
• the second edge is three finite coordinates no larger than six ×1
• the second vector is seven whole numbers under ten ×1
• the second vector is three finite coordinates no larger than six ×1
• the second vector is two finite coordinates no larger than twelve ×1
• the shadow on the x–y plane has the signed area of the product's third component ×1
• the shadow on the y–z plane has the signed area of the product's first component ×1
• the shadow on the z–x plane has the signed area of the product's second component ×1
• the sign of a·b follows the angle ×1
• the squares of the three shadows add to the square of the area itself ×1
• the tangent at each eigenvector's tip runs parallel to the other eigenvector ×1
• the target is three finite coordinates ×1
• the target is two finite coordinates no larger than twelve ×1
• the target's squared length splits into the projection's and the residual's ×1
• the three pairs span three planes with no direction in common ×1
• the two columns are independent, or they span a line rather than a plane ×1
• the two eigenvalues are different, so the directions are determined ×1
• the two vectors are not parallel ×1
• the two vectors are not parallel, or there is nothing left after the subtraction ×1
• the two vectors are not parallel, so there is a plane to write down ×1
• the vector being bounded is two finite coordinates no larger than twelve ×1
• the vector it is measured against is two finite coordinates no larger than twelve ×1
• the vector projected onto is not zero ×1
• the vectors are independent, so nothing vanishes at its own step ×1
• the vectors span a plane ×1
• the weighted product does not care about the order ×1
• the weights are one positive number per point, no larger than a hundred ×1
• there are at least as many coordinates as vectors ×1
• there are more points than coefficients, or nothing is being fitted ×1
• two columns span the plane ×1
• two vectors go in ×1
• under the ordinary product the adjoint is the transpose ×1
• what is left after the subtraction is perpendicular to the first vector ×1

The whole library · What the figures prove