Vector Algebra

Class 12 NCERT

NCERT

1   Represent graphically a displacement of $ 40 \ km, 30^o$ east of north.

Solution :

Here, vector $OP$ represents the displacement of $ 40 km, 30^o$ East of North.

2   Classify the following measures as scalars and vectors.$\\$ $(i) 10 kg $$\\$ $(ii) 2$ meters north-west$\\$$ (iii) 40^o $$\\$$(iv) 40 $ watt$\\$ $(v) 10^{-19}$ coulomb $\\$$(vi) 20 \ m / s ^2$

Solution :

$(i) 10 \ kg $ is a scalar quantity because it involves only magnitude.$\\$ $(ii) 2 $ meters north-west is a vector quantity as it involves both magnitude and direction.$\\$ $(iii) 40^o $ is a scalar quantity as it involves only magnitude.$\\$ $(iv) 40$ watts is a scalar quantity as it involves only magnitude.$\\$ $(v) 10^{-19}$ Coulomb is a scalar quantity as it involves only magnitude.$\\$ $(vi) 20 m / s^2$ is a vector quantity as it involves magnitude as well as direction.

3   Classify the following as scalar and vector quantities.$\\$ (i) time period $\\$(ii) distance$\\$ (iii) force$\\$ (iv) velocity $\\$(v) work done

Solution :

(i) Time period is a scalar quantity as it involves only magnitude. $\\$(ii) Distance is a scalar quantity as it involves only magnitude. $\\$(iii) Force is a vector quantity as it involves both magnitude and direction.$\\$ (iv) Velocity is a vector quantity as it involves both magnitude as well as direction.$\\$ (v) Work done is a scalar quantity as it involves only magnitude.

4   In Figure, identify the following vectors.$\\$ (i) Co-initial$\\$ (ii) Equal$\\$ (iii) Collinear but not equal

Solution :

(i) Vectors $a $ and $d$ are co-initial because they have the same initial point.$\\$ (ii) Vectors $b$ and $d$ are equal because they have the same magnitude and direction.$\\$ (iii) Vectors $a$ and $c$ are collinear but not equal. This is because although they are parallel, their directions are not the same

5   Answer the following as true or false:$\\$ (i) $\overrightarrow{a}$ and $- \overrightarrow{a}$ are collinear.$\\$ (ii) Two collinear vectors are always equal in magnitude.$\\$ (iii)Two vectors having same magnitude are collinear.$\\$ (iv) Two collinear vectors having the same magnitude are equal.

Solution :

(i) True$\\$ $\ \ \ \ \ \ \ $Vectors $\overrightarrow{a}$ and $-\overrightarrow{ a}$ can be parallel or coinciding vectors. Either way the vectors will have same magnitude but opposite in direction and will be parallel to the same line..$\\$ (ii) False$\\$ $\ \ \ \ \ \ \ $Collinear vectors are those vectors that are parallel to the same line.$\\$ (iii) False$\\$ $\ \ \ \ \ \ \ $It is not necessary for two vectors having the same magnitude to be parallel to the same line.$\\$ (iv) False$\\$ $\ \ \ \ \ \ \ $Two vectors are said to be equal if they have the same magnitude and direction, regardless of the positions of their initial points.

6   If a line makes angles $90 ^o ,135 ^o ,45 ^o$ with $x, y$ and $z -$axes respectively, find its direction cosines.

Solution :

Let direction cosines of the line be $l, m,$ and $n.$ $l = \cos 90^o=0\\ m= \cos 135^o =-\dfrac{1}{\sqrt{2}}\\ n = \cos 45^o=\dfrac{1}{\sqrt{2}}$$\\$ Therefore, the direction cosines of the line are $0,-\dfrac{1}{\sqrt{2}} ,$ and $ \dfrac{1}{\sqrt{2}}.$