Unit 07: Vectors

Here is the list of important questions.

  • Find position vector of a point which divide the join of $P$ and $Q$ with position vectors $2\underline i-3 \underline j$ and $3\underline i+2\underline j$ in ratio $4:3$. — BSIC Gujranwala (2016)
  • Find $a$ and $b$ so that the vectors $3\underline i-\underline j+4\underline k$ and $a\underline i+b\underline j+2\underline k$ are parallel. — BSIC Gujranwala (2016)
  • Find $\cos$ of angle between $u.v$ if $u=3\underline i+\underline j-\underline k$, $v=2\underline i-\underline j-\underline k$. — BSIC Gujranwala (2016)
  • Find the unit vector perpendicular to the plane of vectors $a=2\underline i-2\underline j+4\underline k$, $b=-\underline i+\underline j-2\underline k$. — BSIC Gujranwala (2016)
  • Using vectors prove that the points $A(-3,5,-4)$, $B(-1,1,1)$, $C(-1,2,2)$ and $D(-3,4,-5)$ are coplaner. — BSIC Gujranwala (2016)
  • Prove that $\sin (\alpha+\beta)=\sin \alpha \cos \beta+\cos \alpha \sin \beta$. — BSIC Gujranwala (2015), BSIC Sargodha(2017)
  • Prove that $a\times (b+c)+b\times(c+a)+c\times(a+b)=0$. — BSIC Gujranwala (2015)
  • Find the volume of tetrahedron whose vertices are $A(2,1,8)$, $B(3,2,9)$, $C(2,1,4)$ and $D(3,3,10)$. — BSIC Gujranwala (2015)
  • Find the value of $[\underline k \underline i \underline j]$ — BSIC Gujranwala (2015)
  • Use vector method to prove $\sin (\alpha-\beta)=\sin \alpha \cos \beta -\cos \alpha \sin \beta$. — BSIC Gujranwala (2015), FBSIC (2017)
  • Find the angle between the vectors $2i-j+k$ and $-i+j$ — FBSIC (2017)
  • Prove that the altitudes of a triangle are concurrent.— FBSIC (2017)
  • Find a unit vector perpendicular to the plane containing $a$ and $b$, also find sine of angle between them,
    $b=4i+3j-k$.— FBSIC (2016)
  • Find direction cosine of the vector $\underline {r}=x\underline i+y \underline j+zk$.— BSIC Rawalpandi(2017 )
  • Find thevector of length $5$, in the direction opposite to the vector $\underline v =\underline i-2\underline j +3 \underline k$ — BSIC Rawalpandi(2017 )
  • If \underlinea+\underlineb+\underlinec=0 then prove that $\underline a \times \underline b=\underline b\times \underline c=\underline c \times \underline a $.— BSIC Rawalpandi(2017 )
  • Find the value of, so that the vectors $\alpha \underline i+\underline j$, $\underline i+\underline j +3 \underline k$ and $2 \underline i +\underline j -2 \underline k$ are coplanar.— BSIC Rawalpandi(2017 )
  • Prove that, by vector method $\cos (\alpha - \beta)=\cos \alpha \cos \beta +\sin \alpha \sin \beta$.— BSIC Rawalpandi(2017 )
  • Find volume of the tetrahedron whose vertices are $A(2,1,8)$, $B(3,2,9)$, $C(2,1,4)$ and $D(3,3,0)$— BSIC Rawalpandi(2017 )
  • If $\underline u=2\underline i+3\underline j+4\underline k$, $\underline v=4\underline i+6\underline j+2\underline k$, $\underline w=-6\underline i-9 \underline j-3\underline k$, then find $|\underline u-\underline v-\underline w|$.— BSIC Sargodha(2016 )
  • If $\cos \alpha$, $\cos \beta$ and $\cos \gamma$ are direction cosineof a vector, then prove $\cos ^2\alpha+\cos ^2\beta+\cos ^2\gamma=1$— BSIC Sargodha(2016 )
  • Fina a unit vector perpendicular to the plane containing the vectors $\underline a=\underline i+\underline j$, $\underline b=\underline i-\underline j$— BSIC Sargodha(2016 )
  • Find the volume of the parallelepiped determined by $\underline u=\underline i+2\underline j-\underline k$, $\underline v=\underline i-2\underline j+3\underline k$, $\underline w=\underline i-7\underline j-4\underline k$— BSIC Sargodha(2016 )
  • Find volume of the tetrahedron with the vertices $A(0,1,2)$, $B(3,2,1)$, $C(1,2,1)$ and $D(5,5,6)$— BSIC Sargodha(2016 )
  • Find $\alpha$ so that vectors $\underline u=2\alpha \underline i+\underline j-\underline k$ and $\underline v = \underline i+\alpha \underline j +4\underline k$ are perpendicular .— BSIC Sargodha(2017)
  • If $\underline v$ is a vector for which $\underline v.\underline i=0$, $\underline v.\underline j=0$, $\underline v.\underline k=0$. Find $\underline v$.— BSIC Sargodha(2017)
  • Find a unit vector perpendicular to both $\underline a=2\underline i-6\underline j-3\underline k$, $\underline b=4\underline i+3\underline j- \underline k$.— BSIC Sargodha(2017)
  • Prove that $\underline a \times(\underline b+\underline c)+\underline b \times (\underline c+\underline a)+\underline c\times(\underline a+\underline b)=0$— BSIC Sargodha(2017)