Addition of Coplanar Vectors

Excerpts

• Addition of Coplanar Vectors

  By a “vector” is meant a quantity which has magnitude and direction.

• Addition of Coplanar Vectors

  Though a vector is represented by a line, its physical dimensions may be different from that of a line.

• Addition of Coplanar Vectors

  The diagonal $OC$ represents in magnitude and direction and point of application the resultant of $OA$ and $OB$.

• Addition of Coplanar Vectors

  The composition of successive vectors partakes more of the nature of multiplication than of addition.

• Addition of Coplanar Vectors

  The area between the curve $OPQ$ and the vector $OQ$ depends on the path, and has a physical meaning.

• Addition of Coplanar Vectors

  In the case of a sum of simultaneous vectors applied at a common point, the ordinary rule about the transposition of a term in an equation holds good.

• Addition of Coplanar Vectors

  “Reverse the direction of the component, then compound it with the given resultant to find the required component.”

• Addition of Coplanar Vectors

  It is a mistake to attempt to found space-analysis upon arbitrary formal laws; the fundamental rules must be made to express universal properties of the thing denoted.

Equations

• Addition of Coplanar Vectors

  B = b\beta

  A vector B is written as its magnitude b times its direction β, which are kept as separate letters.

• Addition of Coplanar Vectors

  f^2 = f_1^2 + f_2^2 + 2f_1f_2 \cos \theta_2

  The square of the magnitude of the resultant of two simultaneous components is the sum of their squares plus twice their product times the cosine of the angle between them.

• Addition of Coplanar Vectors

  \tan \theta =\frac{f_2\sin\theta_2}{f_1 + f_2\cos\theta_2}

  The tangent of the direction angle of the resultant equals the sine of the second component's angle times its magnitude, divided by the first magnitude plus the second's cosine term.

• Addition of Coplanar Vectors

  OC = \sqrt{f_1^2 + f_2^2 + 2f_1f_2\cos\theta_2} \underline{\left/\tan^{-1} \frac{f_2\sin \theta_2}{f_1 + f_2\cos\theta_2}\right.}

  The resultant OC of two simultaneous vectors, written in magnitude and direction form, has magnitude given by the law of cosines and direction given by the arctangent of the ratio above.

• Addition of Coplanar Vectors

  OC = 2f_1\cos\frac{\theta_2}{2} \underline{\left/\frac{\theta_2}{2}\right.}

  When two simultaneous components are equal in magnitude, the resultant lies along the bisector of the angle between them and has magnitude twice the projection of either component on that bisector.

• Addition of Coplanar Vectors

  f_2\underline{/\theta_2} = \sqrt{f^2 + f_1^2 - 2ff_1\cos\theta} \underline{\left/\tan^{-1} \frac{f\sin\theta}{-f_1 + f\cos\theta}\right.}

  Given a vector and one component, the other component is the vector difference of the given vector and the given component, written in magnitude and direction form.

• Addition of Coplanar Vectors

  f_1 + f_2\cos(\theta_2 - \theta_1) = f\cos(\theta - \theta_1)

  The sum of the first component and the projection of the second component on the direction of the first equals the projection of the resultant on that same direction.

• Addition of Coplanar Vectors

  f_1\cos(\theta_2 - \theta_1) + f_2 = f\cos(\theta_2 - \theta)

  The projection of the first component on the direction of the second, plus the second component, equals the projection of the resultant on the second component's direction.

• Addition of Coplanar Vectors

  f_1 = f\frac{ \{\cos(\theta - \theta_1) - \cos(\theta_2 - \theta)\cos(\theta_2 - \theta_1)\} } {1 - \cos^2(\theta_2 - \theta_1)}

  Solving the two projection equations gives the magnitude of the first component when the resultant and the directions of both components are given.

• Addition of Coplanar Vectors

  \sqrt{\left( \sum f\cos\theta \right)^2 + \left( \sum f\sin\theta \right )^2} \cdot \tan^{-1}\frac{\sum f\sin\theta}{\sum f\cos\theta}

  The resultant of any number of simultaneous vectors has magnitude equal to the square root of the sum of squared cosine-sums and sine-sums, and direction equal to the arctangent of the sine-sum over the cosine-sum. Note: the chapter's intermediate line before this result reads \sum f\sum\theta where \sum f\sin\theta is expected; this is recorded as a possible erratum and was not corrected.

• Addition of Coplanar Vectors

  A + B = -C

  For simultaneous vectors with no order of succession, a term may be transposed across an equation by reversing its sign, so that A + B equals minus C when A + B + C is zero.

Problems