\displaystyle \textbf{1.}\ \text{A vector is a physical quantity having both magnitude and direction.}

\displaystyle \textbf{2.}\ \text{If }\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\text{ are the vectors represented by the sides of a triangle taken in order, then}
\displaystyle \overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{0}.
\displaystyle \text{Conversely, if }\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\text{ are three non-collinear vectors, such that}
\displaystyle \overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{0},\text{ then they form the sides of a triangle taken in order.}

\displaystyle \textbf{3.}\ \text{Two non-zero vectors }\overrightarrow{a}\text{ and }\overrightarrow{b}\text{ are collinear iff there exist non-zero scalars }x\text{ and }y\text{ such that}
\displaystyle x\overrightarrow{a}+y\overrightarrow{b}=\overrightarrow{0}.

\displaystyle \textbf{4.}\ \text{If }\overrightarrow{a}\text{ and }\overrightarrow{b}\text{ are two non-zero non-collinear vectors, then }x\overrightarrow{a}+y\overrightarrow{b}=\overrightarrow{0}\Rightarrow x=y=0.

\displaystyle \textbf{5.}\ \text{If }\overrightarrow{a}\text{ and }\overrightarrow{b}\text{ are two non-zero vectors, then any vector }\overrightarrow{r}\text{ coplanar with }\overrightarrow{a}\text{ and }\overrightarrow{b}\text{ can be} \\ \text{uniquely expressed as }\overrightarrow{r}=x\overrightarrow{a}+y\overrightarrow{b},\text{ where }x,y\text{ are scalars.}
\displaystyle \text{Also, }\overrightarrow{r}=\left\{x\left|\overrightarrow{a}\right|\right\}\widehat{a}+\left\{y\left|\overrightarrow{b}\right|\right\}\widehat{b}

\displaystyle \textbf{6.}\ \text{If }\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\text{ are three given non-coplanar vectors, then every vector }\overrightarrow{r}\text{ in space can be} \\ \text{uniquely expressed as }\overrightarrow{r}=x\overrightarrow{a}+y\overrightarrow{b}+z\overrightarrow{c}\text{ for some scalars }x,y,z.
\displaystyle \text{or, }\overrightarrow{r}=\left\{x\left|\overrightarrow{a}\right|\right\}\widehat{a}+\left\{y\left|\overrightarrow{b}\right|\right\}\widehat{b}+\left\{z\left|\overrightarrow{c}\right|\right\}\widehat{c}
\displaystyle \text{Here, vectors }x\overrightarrow{a},y\overrightarrow{b}\text{ and }z\overrightarrow{c}\text{ are called the components of }\overrightarrow{r}\text{ in the directions of }\overrightarrow{a},\overrightarrow{b},\overrightarrow{c} \\ \text{respectively and the scalars }x,y,z\text{ are known as the coordinates of }\overrightarrow{r}\text{ relative to the triad of} \\ \text{non-coplanar vectors }\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}.
\displaystyle \text{The triad of non-coplanar vectors }\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\text{ relative to which we decompose any vector }\\ \overrightarrow{r}\text{ is called a base.}
\displaystyle \text{The scalars }x\left|\overrightarrow{a}\right|,y\left|\overrightarrow{b}\right|,z\left|\overrightarrow{c}\right|\text{ are known as the projections of }\overrightarrow{r}\text{ in the directions of } \\ \overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\text{ respectively.}

\displaystyle \textbf{7.}\ \text{If }\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\text{ are three non-zero non-coplanar vectors and }x,y,z\text{ are three scalars, then } \\ x\overrightarrow{a}+y\overrightarrow{b}+z\overrightarrow{c}=\overrightarrow{0}\Rightarrow x=y=z=0.

\displaystyle \textbf{8.}\ \text{If }l,m,n\text{ are direction cosines of a vector }\overrightarrow{r}(=\overrightarrow{OP}),\text{ where }O\text{ is the origin and} \\ \text{the point }P\text{ has }(x,y,z)\text{ as }\text{its coordinates, then}
\displaystyle (i)\ l^{2}+m^{2}+n^{2}=1
\displaystyle (ii)\ x=l\left|\overrightarrow{r}\right|,\ y=m\left|\overrightarrow{r}\right|,\ z=n\left|\overrightarrow{r}\right|
\displaystyle (iii)\ \overrightarrow{r}=\left|\overrightarrow{r}\right|\left(l\widehat{i}+m\widehat{j}+n\widehat{k}\right)
\displaystyle (iv)\ \overrightarrow{r}=l\widehat{i}+m\widehat{j}+n\widehat{k}

\displaystyle \textbf{9.}\ \text{If }\overrightarrow{r}=a\widehat{i}+b\widehat{j}+c\widehat{k},\text{ then }a,b,c\text{ are proportional to its direction ratios and its direction cosines are}
\displaystyle \frac{\pm a}{\sqrt{a^{2}+b^{2}+c^{2}}},\ \frac{\pm b}{\sqrt{a^{2}+b^{2}+c^{2}}},\ \frac{\pm c}{\sqrt{a^{2}+b^{2}+c^{2}}}

\displaystyle \textbf{10.}\ (i)\ \text{A set of non-zero vectors }\overrightarrow{a}_{1},\overrightarrow{a}_{2},\overrightarrow{a}_{3},\ldots,\overrightarrow{a}_{n}\text{ is linearly} \\ \text{independent, if }x_{1}\overrightarrow{a}_{1}+x_{2}\overrightarrow{a}_{2}+\cdots+x_{n}\overrightarrow{a}_{n}=\overrightarrow{0}\Rightarrow x_{1}=x_{2}=\cdots=x_{n}=0
\displaystyle (ii)\ \text{A set of vectors }\overrightarrow{a}_{1},\overrightarrow{a}_{2},\overrightarrow{a}_{3},\ldots,\overrightarrow{a}_{n}\text{ is linearly dependent, if there} \\ \text{exist scalars }x_{1},x_{2},\ldots,x_{n}\text{ not} \text{all zero such that }x_{1}\overrightarrow{a}_{1}+x_{2}\overrightarrow{a}_{2}+\cdots+x_{n}\overrightarrow{a}_{n}=\overrightarrow{0}.
\displaystyle (iii)\ \text{Any two non-zero, non-collinear vectors are linearly independent.}
\displaystyle (iv)\ \text{Any two collinear vectors are linearly dependent.}
\displaystyle (v)\ \text{Any three non-coplanar vectors are linearly independent.}
\displaystyle (vi)\ \text{Any three coplanar vectors are linearly dependent.}
\displaystyle (vii)\ \text{Any set of four or more vectors in three dimensional space is linearly dependent set.}

\displaystyle \textbf{11.}\ \text{If }A\text{ and }B\text{ are two points with position vectors }\overrightarrow{a}\text{ and }\overrightarrow{b}\text{ respectively, then the} \\ \text{position vector of a point }C\text{ dividing }AB\text{ in the ratio }m:n\text{ internally and externally are}
\displaystyle \frac{m\overrightarrow{b}+n\overrightarrow{a}}{m+n}\text{ and }\frac{m\overrightarrow{b}-n\overrightarrow{a}}{m-n}\text{ respectively.}

\displaystyle \textbf{12.}\ \text{If }A\text{ and }B\text{ are two points with position vectors }\overrightarrow{a}\text{ and }\overrightarrow{b} \text{ respectively }\text{and }m,n \\ \text{ are positive real numbers, then }m\overrightarrow{OA}+n\overrightarrow{OB}=(m+n)\overrightarrow{OC},\text{ where }C\text{ is a point on } \\ AB\text{ dividing it in the ratio }n:m.
\displaystyle \text{Also, }\overrightarrow{OA}+\overrightarrow{OB}=2\overrightarrow{OC},\text{ where }C\text{ is the mid-point of }AB.

\displaystyle \textbf{13.}\ \text{If }S\text{ is any point in the plane of a triangle }ABC,\text{ then }\overrightarrow{SA}+\overrightarrow{SB}+\overrightarrow{SC}=3\overrightarrow{SG},\text{ where }G\text{ is the centroid of }\triangle ABC.

\displaystyle \textbf{14.}\ \text{The necessary and sufficient condition for three points with position vectors } \\ \overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\text{ to be collinear is that there exist scalars }x,y,z\text{ not all zero such that } \\ x\overrightarrow{a}+y\overrightarrow{b}+z\overrightarrow{c}=\overrightarrow{0},\text{ where }x+y+z=0.

\displaystyle \textbf{15.}\ \text{The necessary and sufficient condition for four points with position vectors } \\ \overrightarrow{a},\overrightarrow{b},\overrightarrow{c},\overrightarrow{d}\text{ to be coplanar is that there exist scalars }x,y,z,t\text{ not all} \\ \text{zero such that }x\overrightarrow{a}+y\overrightarrow{b}+z\overrightarrow{c}+t\overrightarrow{d}=\overrightarrow{0},\text{ where }x+y+z+t=0.

Discover more from ICSE / ISC / CBSE Mathematics Portal for K12 Students

Subscribe to get the latest posts sent to your email.