Vector Solutions, Shortcuts, Formulas

Points to Remember

  • position vector of a point p(x,y,z) is given as \vec{OP}(\vec{r})=x\hat{i}+y\hat{j}+z\hat{k} , and it’s magnitude by \sqrt{x^{2}+y^{2}+z^{2}}
  • The scalar components of a vector are its direction ratios, and represent its projections along the respective axes.
  • The magnitude (r ), direction ratios (a , b, c) and direction cosines (l, m, n) of any vector are related as l=\frac{a}{r},m\frac{b}{r},n=\frac{c}{r}
  • Th e vector sum of the three sides of a triangle taken in order is 0
  • The vector sum of two coinitial vectors is given by the diagonal of the parallelogram whose adjacent sides are the given vectors.
  • The multiplication of a given vector by a scalar λ , changes the magnitude of the vector by the multiple |λ|, and keeps the direction same (or makes it opposite) according as the value of λ is positive (or negative).
  • For a given a , the vector \hat{a}=\frac{\vec{a}}{|\vec{a}|} gives the unit vector in the direction of vector a.
  • The position vector of a point R dividing a line segment joining the points P and Q whose position vectors are \vec{a}\:and\:\vec{b} respectively in the ratio m: nInternally is given by : \frac{n\vec{a}+m\vec{b}}{m+n} Externally , is given by : \frac{n\vec{b}-m\vec{a}}{m+n}
  • The scalar product of two given vector \vec{a}\:and\:\vec{b} having angle θ between them is define as \vec{a}\cdot \vec{b}=|\vec{a}||\vec{b}|\cos \theta Also when \vec{a}\cdot \vec{b} is given the angle , θ between the vector \vec{a}\:and\:\vec{b} may be determine by \cos \theta =\frac{\vec{a}\cdot \vec{b}}{|\vec{a}||\vec{b}|}
  • If θ is the angle between two vectors \vec{a}\:and\:\vec{b} , then their cross product is given as \vec{a}\times \vec{b}=|\vec{a}||\vec{b}|\sin \theta \hat{n} , where unit vector \hat{n} , perpendicular to the plane containing \vec{a}\:and\:\vec{b} . Such that \vec{a},\vec{b}, hat{n} . form right handed system of coordinate axes.

 

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