Product of Two Vectors

Excerpts

• Product of Two Vectors

  The formula $\text{velocity flux} = \text{electromotive-force}$ is much handier than any thumb-and-finger rule; for it compares the three directions directly with the right-handed screw.

• Product of Two Vectors

  The square combinations give results which are independent of direction, and consequently are summed by simple addition.

• This excerpt is in Products of Coplanar Vectors (Products of Coplanar Vectors)

• Product of Two Vectors

  The vector product as before is denoted by $\mathrm{V}AB$. It means the product of $A$ and the component of $B$ which is perpendicular to $A$, and is represented by the area of the parallelogram formed by $A$ and $B$.

• Product of Two Vectors

  The product is positive when the vector and the projection have the same direction, and negative when they have opposite directions.

• Product of Two Vectors

  Frequently all that is demanded is, given two of these directions to determine the third.

• Product of Two Vectors

  In a sum of vectors, the vectors are necessarily homogeneous, but in a product the vectors may be heterogeneous.

Equations

• Product of Two Vectors

  ij &= k, & jk &= i, & ki &= j

  The right-handed cyclic rules for products of the unit vectors in space: i times j is k, j times k is i, and k times i is j, taken as assumed by symmetry of space.

• Product of Two Vectors

  \text{velocity flux} = \text{electromotive-force}

  In the dynamo, the velocity of the conductor combined with the magnetic flux with the right-handed screw rule gives the direction of the electromotive force. The chapter states it in words, so no letter symbols are defined.

• Product of Two Vectors

  \text{current flux} = \text{mechanical-force}

  In the electric motor, the current in the conductor combined with the magnetic flux gives the direction of the mechanical force on the conductor. The chapter states it in words, so no letter symbols are defined.

• Product of Two Vectors

  \text{flux force} = \text{current}

  Cyclical permutation of the motor formula gives flux and force producing current, stated in words.

• Product of Two Vectors

  A &= a_1j + a_2j + a_3k

  Defines the vector A by its three components along the axes. The chapter prints a_1j for the first term where a_1i is evidently meant; this is a typesetting slip in the source, flagged here and not corrected.

• Product of Two Vectors

  a_1b_1 + a_2b_2 + a_3b_3 =\mathrm{S}AB

  The scalar product of A and B is the sum of the products of their corresponding components; it is independent of direction.

• This equation is in Products of Coplanar Vectors (Products of Coplanar Vectors)

• Product of Two Vectors

  A^2 = {a_1}^2 + {a_2}^2 + {a_3}^2 = a^2

  The square of a vector is the sum of the squares of its components, which equals the square of its magnitude a, and is always positive.

• This equation is in Products of Coplanar Vectors (Products of Coplanar Vectors)
• This equation is in Products of Coplanar Vectors (Products of Coplanar Vectors)
• This equation is in Products of Coplanar Vectors (Products of Coplanar Vectors)

• Product of Two Vectors

  \mathrm{S}SB = b_ls_l + b_2s_2 + b_3s_3

  The magnetic flux through an area S is the scalar product of the flux intensity B with the area vector S. Flagged: the chapter prints b_l s_l, presumably b_1 s_1, and the operator as SSB, read here as S applied to S and B.

• Product of Two Vectors

  (A + B)C = AC + BC

  The vector product distributes over a sum of vectors in the first factor: the product of the sum A+B with C equals AC plus BC.

• Product of Two Vectors

  (A+B)(C+D) = AC + AD + BC + BD

  The product of two sums of non-successive vectors expands term by term, each sum distributing over the other.

• Product of Two Vectors

  (A+B)^2 & = A^2 + B^2 + AB + BA

  The square of a sum of vectors expands to the squares of each vector plus the product of A and B and the product of B and A.

• Product of Two Vectors

  A^2 + B^2 + 2\mathrm{S}AB

  For a sum of vectors, the square equals the sum of the squares of the two vectors plus twice their scalar product.

Problems