Login/Sign Up
Home » If and , then match the relations in column I with the angle between between and in column II.
If |A|=2 and |B|=4, then match the relations in column I with the angle between \theta between A and B in column II.
CBSE | Class 11 | Motion in a Plane | NCERT
If |A|=2 and |B|=4, then match the relations in column I with the angle between \theta between A and B in column II.
Column I Column II
a) A.B = 0 i) θ = 0
b) A.B = +8 ii) θ = 90o
c) A.B = 4 iii) θ = 180o
d) A.B = -8 iv) θ = 0o

 

Answer:

a) matches with ii)

b) matches with i)

c) matches with iv)

d) matches with iii)

As we know that \overrightarrow{ A } \cdot \overrightarrow{ B }=|\overrightarrow{ A } \| \overrightarrow{ B }| \cos \theta

    \[\begin{aligned} &\text { (a) } \overrightarrow{ A } \cdot \overrightarrow{ B }=|\overrightarrow{ A } \| \overrightarrow{ B }| \cos \theta=0 \\ &2 \times 4 \cos \theta=0 \\ &\cos \theta=\cos 90^{\circ} \end{aligned}\]

Thus, Option (a) of column I matches with option (ii) in column II.

(b) Given, \overrightarrow{ A } \cdot \overrightarrow{ B }=8
Implying the formula, |\overrightarrow{ A }||\overrightarrow{ B }| \cos \theta=8
2 \times 4 \cos \theta=8
\therefore \cos \theta=1=\cos 0^{\circ}
Solving we get, \theta=0^{\circ}
So option (b) of column I matches with option (i) of column II.

    \[\begin{aligned} &\text { (c) } \overrightarrow{ A } \cdot \overrightarrow{ B }=|\overrightarrow{ A } \| \overrightarrow{ B }| \cos \theta=2 \times 4 \cos \theta \\ &=8 \cos \theta \end{aligned}\]

8 \cos \theta=4 as in option (c) of column I.

cosθ=48=12⟹cosθ=cos60°⟹θ=60°
\cos \theta=\frac{4}{8}=\frac{1}{2} \Longrightarrow \cos \theta=\cos 60^{\circ} \Longrightarrow \theta=60^{\circ}

Hence, option (c) of column I matches with option (iv) of column II.

Solving using formula.

    \[\begin{aligned} &\text { (d) } \overrightarrow{ A } \cdot \overrightarrow{ B }=-8 \\ &|\overrightarrow{ A }||\overrightarrow{ B }| \cos \theta=-8 \\ &2 \times 4 \cos \theta=-8 \\ &\cos \theta=-1 \Longrightarrow \cos \theta=\cos 180^{\circ} \Longrightarrow \theta=180^{\circ} \end{aligned}\]

As a result, option (d) in column I corresponds to option (iii) in column II.

 

i

More Material