[커세라] 이과생을 위한 벡터 미적분
W1-3: 2강. 직교좌표계 (Cartesian Coordinates)
방향(한개 이상의 좌표축)과 크기(좌표의 눈금)를 가진 벡터(Vector)를 표현하려면 좌표계가 필요하다. 먼저 가장 직관적인 좌표계는 직교 좌표계다.
1. 직교좌표계: 공간을 나타내는 좌표축이 직교(orthogonal)하고 단위벡터의 길이가 1인(norm) 좌표계(ortho-normal)
2. 직교 좌표계의 단위벡터: 3축(차원) 직교좌표계의 단위벡터(unit vector)를 각각 i, j, k 라 하자.
* 단위 벡터(unit vector):
- 스칼라(값) 1에 방향을 부여함으로써 벡터화 한 것.
- 벡터에 벡터 크기(값)을 나누면 단위벡터가 된다.
3. 벡터 A의 표현: 벡터는 성분(값)과 방향으로 표현된다. 3차원 직교 좌표계에서 벡터 A는 좌표축의 성분과 단위벡터 곱의 합이다.
4. 벡터 A 의 크기는 각 좌표축 성분 제곱을 모두 더한 값의 제곱근(직교좌표계)
5. 위치 벡터(Position Vector): 좌표축 성분에 단위 벡터를 곱한 벡터를 모두 더한 것
6. 변위 벡터(Displacement Vector): 두 벡터의 차분 벡터.
두 벡터의 차분으로 구해진 변위벡터는 원점에 구애 되지 않는다. 벡터로 기술된 객체는 좌표계에 무관하다.
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Reading: Newton's equation for the force between two masses
(a) Given a Cartesian coordinate system with standard unit vectors i, j, and k, let the mass m_1 be at position r1=x1i+y1j+z1k and the mass m_2 be at position r2=x2i+y2j+z2k. In terms of the standard unit vectors, determine the unit vector that points from m_1 to m_2.
(b) Newton's law of universal gravitation states that two point masses attract each other along the line connecting them, with a force proportional to the product of their masses and inversely proportional to the square of the distance between them. The magnitude of the force acting on each mass is therefore
$$
\displaystyle F = G\frac{m_1m_2}{r^2}
$$
where m_1 and m_2 are the two masses, r is the distance between them, and G is the gravitational constant. Let the masses m_1 and m_2 be located at the position vectors r1 and r2. Write down the vector form for the force acting on m_1 due to its gravitational attraction to m_2.
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[목차][이전][다음]
(a) Given a Cartesian coordinate system with standard unit vectors i, j, and k, let the mass m_1 be at position r1=x1i+y1j+z1k and the mass m_2 be at position r2=x2i+y2j+z2k. In terms of the standard unit vectors, determine the unit vector that points from m_1 to m_2.
(b) Newton's law of universal gravitation states that two point masses attract each other along the line connecting them, with a force proportional to the product of their masses and inversely proportional to the square of the distance between them. The magnitude of the force acting on each mass is therefore
$$
\displaystyle F = G\frac{m_1m_2}{r^2}
$$
where m_1 and m_2 are the two masses, r is the distance between them, and G is the gravitational constant. Let the masses m_1 and m_2 be located at the position vectors r1 and r2. Write down the vector form for the force acting on m_1 due to its gravitational attraction to m_2.
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[목차][이전][다음]
