Addition of Vectors in Space

Excerpts

• Addition of Vectors in Space

  A vector in space can be expressed in terms of three independent components, and when these form a rectangular set the directions of resolution are expressed by $i$, $j$, $k$.

• Addition of Vectors in Space

  In space the symbol $\rho$ for the direction involves two elements.

• Addition of Vectors in Space

  The additional angle $\overline{\phi/}$ is introduced to specify the plane in which the angle from the initial line lies.

• Addition of Vectors in Space

  When the successive vectors do not lie in one plane, the several elements of the area enclosed will lie in different planes, but these add by vector addition into a resultant directed area.

Equations

• Addition of Vectors in Space

  R = r\rho = xi + yj + zk

  A variable vector R equals its magnitude r times its direction rho, and also equals its rectangular components x, y, z along the unit vectors i, j, k.

• Addition of Vectors in Space

  B = b\beta = b_1i + b_2j + b_3k

  A constant vector B equals its magnitude b times its direction beta, and also equals its components b_1, b_2, b_3 along i, j, k.

• Addition of Vectors in Space

  \rho = \frac{xi + yj + zk}{x^2 + y^2 + z^2}

  The direction rho of a vector with components x, y, z is written as those components divided by the sum of their squares, which is valid because the book takes that sum to be unity.

• Addition of Vectors in Space

  R = r \cos\theta \cdot i + r \sin\theta \cos \phi \cdot j + r \sin\theta \sin\phi \cdot k

  A vector given by magnitude r and the two angles phi and theta has components r cos theta along i, r sin theta cos phi along j, and r sin theta sin phi along k.

• Addition of Vectors in Space

  R = \sqrt{x^2 + y^2 + z^2}\ \overline{\left.\tan^{-1}\frac{z}{y}\right/}\!\!\!\! \underline{\left/\tan^{-1}\frac{\sqrt{y^2 + z^2}}{x}\right.}

  A vector with components x, y, z has magnitude equal to the square root of the sum of their squares, and its two angles are given by inverse tangents of z over y and of the root of y squared plus z squared over x.

• Addition of Vectors in Space

  \sum R = \left(\sum x \right)i + \left(\sum y \right)j + \left(\sum z \right)k

  The resultant of several vectors is found by summing their x, y and z components separately and combining them along i, j, k.

• Addition of Vectors in Space

  r = \sqrt{\left(\sum x \right)^2 + \left(\sum y \right)^2 + \left(\sum z \right)^2}

  The magnitude of the resultant is the square root of the sum of the squares of the summed components.

• Addition of Vectors in Space

  \tan\phi = \frac{\sum z}{\sum y}

  The tangent of the angle phi of the resultant equals the summed z component divided by the summed y component.

• Addition of Vectors in Space

  \tan\theta = \frac{\sqrt{\left(\sum y \right)^2 + \left(\sum z \right)^2}}{\sum x}

  The tangent of the angle theta of the resultant equals the root of the summed y and z components squared, divided by the summed x component.

Problems