# 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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