Direction cosines

Japanese: 方向余弦 - ほうこうよげん

The components of a unit vector in a given direction with respect to an orthonormal basis are called the direction cosines of that direction. In other words, when a non-zero vector a is given on a plane, the components (λ, μ) of a/|a| with respect to the orthogonal coordinate system are called the direction cosines of a. Here, |a| represents the length of a. Therefore, a/|a| represents a unit vector in the a direction. If the angles that a makes with the x-axis and y-axis are α and β, respectively, then λ=cosα, μ=cosβ
where λ ² +μ ² =1 is satisfied ( Figure A ). In addition, when a non-zero vector a is given in space, the components (λ,μ,ν) of a/‖a‖ in the Cartesian coordinate system are called the direction cosines of a. If the angles that a makes with the x-axis, y-axis, and z-axis are α, β, and γ, respectively, then λ=cosα, μ=cosβ, ν=cosγ
and λ ² +μ ² +ν ² =1 holds. Hesse's standard form for a line in the plane λx+μy=p
or the Hesse normal form for a plane of space λx+μy+νz=p
In (λ,μ),(λ,μ,ν) are nothing but the direction cosines of the normal ( Figure B ).

[Koichi Ogiue]

Direction cosine explanation diagram (Figure A)
©Shogakukan ">

Direction cosine explanation diagram (Figure A)

Direction cosine explanation diagram (Figure B)
©Shogakukan ">

Direction cosine explanation diagram (Figure B)

Source: Shogakukan Encyclopedia Nipponica About Encyclopedia Nipponica Information | Legend

Japanese:

与えられた方向の単位ベクトルの正規直交基底に関する成分を、その方向の方向余弦という。すなわち、平面上にゼロでないベクトルaが与えられたとき、直交座標系に関するa/‖a‖の成分（λ,μ）をaの方向余弦という。ここで、‖a‖は、aの長さを表す。したがってa/‖a‖は、a方向の単位ベクトルを表す。aがx軸、y軸となす角をそれぞれα、βとすれば
　　λ=cosα, μ=cosβ
でありλ²+μ²=1を満たす（図A）。また、空間にゼロでないベクトルaが与えられたとき、直交座標系に関するa/‖a‖の成分（λ,μ,ν）をaの方向余弦という。aがx軸、y軸、z軸となす角をそれぞれα、β、γとすれば
　　λ=cosα, μ=cosβ, ν=cosγ
でありλ²+μ²+ν²=1が成り立つ。平面上の直線に対するヘッセの標準形
　　λx+μy=p
や、空間の平面に対するヘッセの標準形
　　λx+μy+νz=p
における(λ,μ),(λ,μ,ν)は法線方向の方向余弦にほかならない（図B）。