Class 12: Algebra of Vectors

$\displaystyle \textbf{Question 1: }~\text{Represent the following graphically:}\\ \text{(i) a displacement of }40\ \text{km, }30^{\circ}\text{ east of north}\\ \text{(ii) a displacement of }50\ \text{km south-east}\\ \text{(iii) a displacement of }70\ \text{km, }40^{\circ}\text{ north of west.}$
$\displaystyle \text{Answer:}$

$\displaystyle \textbf{Question 2: }~\text{Classify the following measures as scalars and vectors:}\\ \text{(i) }15\ \text{kg}\qquad \text{(ii) }20\ \text{kg weight}\qquad \text{(iii) }45^{\circ}\\ \text{(iv) }10\ \text{meters south-east}\qquad \text{(v) }50\ \text{m/sec}^{2}.$
$\displaystyle \text{Answer:}$
$\displaystyle \text{The quantities which have only magnitude and which are not related to any} \\ \text{fixed direction in space are called scalar quantities or simply scalars.}$
$\displaystyle \text{The quantities which have both magnitude and direction are called vector} \\ \text{quantities or simply vectors.}$
$\displaystyle \text{(i) Mass -- Scalar}$
$\displaystyle \text{(ii) Weight (Force) -- Vector}$
$\displaystyle \text{(iii) Angle -- Scalar}$
$\displaystyle \text{(iv) Directed Distance -- Vector}$
$\displaystyle \text{(v) Magnitude of Acceleration -- Scalar}$

$\displaystyle \textbf{Question 3: }~\text{Classify the following as scalar and vector quantities:}\\ \text{(i) Time period}\qquad \text{(ii) Distance}\qquad \text{(iii) Displacement}\\ \text{(iv) Force}\qquad \text{(v) Work}\qquad \text{(vi) Velocity}\qquad \text{(vii) Acceleration.}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{The quantities which have only magnitude and which are not related to any fixed} \\ \text{direction in space are called scalar quantities or simply scalars.}$
$\displaystyle \text{The quantities which have both magnitude and direction are called vector quantities} \\ \text{or simply vectors.}$
$\displaystyle \text{(i) Scalar}$ $\displaystyle \text{(ii) Scalar}$ $\displaystyle \text{(iii) Vector}$ $\displaystyle \text{(iv) Vector}$ $\displaystyle \text{(v) Scalar}$ $\displaystyle \text{(vi) Vector}$ $\displaystyle \text{(vii) Vector}$

$\displaystyle \textbf{Question 4: }~\text{In Fig. }ABCD\text{ is a regular hexagon, which vectors are:}\\ \text{(i) Collinear}\qquad \text{(ii) Equal}\qquad \text{(iii) Coinitial}\qquad \text{(iv) Collinear but not equal.}$

$\displaystyle \text{Answer:}$
$\displaystyle \text{(i) Vectors having the same or parallel supports are called collinear vectors.}$
$\displaystyle \text{In the given figure, the collinear vectors are } \vec{a},\vec{d};\vec{x},\vec{z};\vec{b},\vec{c},\vec{y}$
$\displaystyle \text{(ii) Vectors having the same magnitude and direction are called equal vectors.}$
$\displaystyle \text{In the given figure, the equal vectors are } \vec{b},\vec{x};\vec{c},\vec{y};\vec{a},\vec{d}$
$\displaystyle \text{(iii) Vectors having the same initial point are called co-initial vectors.}$
$\displaystyle \text{In the given figure, the co-initial vectors are } \vec{a},\vec{y},\vec{z}; \vec{x},\vec{d}$
$\displaystyle \text{(iv) The vectors which are collinear but not equal are } \vec{b},\vec{z};\vec{x},\vec{z}$

$\displaystyle \textbf{Question 5: }~\text{Answer the following as true or false:}\\ \text{(i) }\vec{a}\text{ and }-\vec{a}\text{ are collinear.}\\ \text{(ii) Two collinear vectors are always equal in magnitude.}\\ \text{(iii) Zero vector is unique.}\\ \text{(iv) Two vectors having same magnitude are collinear.}\\ \text{(v) Two collinear vectors having the same magnitude are equal.}$
$\displaystyle \text{Answer:}$
$\displaystyle (i)\ \text{True, }\vec a \text{ and } -\vec a \text{ are collinear.}$
$\displaystyle (ii)\ \text{False, Collinear vectors only mean they are along the same or opposite direction,} \\ \text{but their lengths (magnitudes) can be different.}$
$\displaystyle (iii)\ \text{True, The zero vector has zero magnitude and no specific direction, and there is} \\ \text{only one such vector in a given vector space.}$
$\displaystyle (iv)\ \text{False, Vectors can have the same length but be in different directions, so} \\ \text{equal magnitude does not imply collinearity.}$
$\displaystyle (v)\ \text{False, Even if two vectors are collinear and have the same magnitude, they } \\ \text{may be in opposite directions.Only if they are in the same direction } \\ \text{will they be equal.}$