CUET Mathematics 2024 - If a, b and c are three vectors such that a+b+c=0, where a and b are unit vectors and |c|=2, then the angle between the vectors b and c is : | PYQs + Solutions

CUET Mathematics 2024

Algebra

Vector Algebra

Medium

If $\vec{a}, \vec{b}$ and $\vec{c}$ are three vectors such that $\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}$ , where $\vec{a}$ and $\vec{b}$ are unit vectors and $|\vec{c}|=2$ , then the angle between the vectors $\vec{b}$ and $\vec{c}$ is :

60^{\circ}

90^{\circ}

120^{\circ}

180^{\circ}

✅ Correct Option: 4

Given:

\newline

\vec{a} + \vec{b} + \vec{c} = \vec{0}

\newline

|\vec{a}| = |\vec{b}| = 1

(unit vectors)

\newline

|\vec{c}| = 2

Rearranging the equation:

\vec{c} = -\vec{a} - \vec{b}

The angle between vectors can be found using the dot product formula:

\newline

\vec{b} \cdot \vec{c} = |\vec{b}||\vec{c}|\cos\theta

Where

\theta

is the angle between the vectors.

Calculating

\vec{b} \cdot \vec{c}

\newline

\vec{b} \cdot \vec{c} = \vec{b} \cdot (-\vec{a} - \vec{b}) = -\vec{b} \cdot \vec{a} - \vec{b} \cdot \vec{b}

Since

\vec{b}

is a unit vector:

\vec{b} \cdot \vec{b} = |\vec{b}|^2 = 1

So:

\vec{b} \cdot \vec{c} = -\vec{b} \cdot \vec{a} - 1

Using the magnitude of

\vec{c}

\newline

|\vec{c}|^2 = \vec{c} \cdot \vec{c} = (-\vec{a} - \vec{b}) \cdot (-\vec{a} - \vec{b})

\newline

= \vec{a} \cdot \vec{a} + 2\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{b}

\newline

= |\vec{a}|^2 + 2\vec{a} \cdot \vec{b} + |\vec{b}|^2

\newline

= 1 + 2\vec{a} \cdot \vec{b} + 1

\newline

= 2 + 2\vec{a} \cdot \vec{b}

Since

|\vec{c}|^2 = 4

\newline

4 = 2 + 2\vec{a} \cdot \vec{b}

\newline

2\vec{a} \cdot \vec{b} = 2

\newline

\vec{a} \cdot \vec{b} = 1

Calculating final dot product:

\newline

\vec{b} \cdot \vec{c} = -\vec{a} \cdot \vec{b} - 1 = -1 - 1 = -2

Finding the angle:

\newline

\vec{b} \cdot \vec{c} = |\vec{b}||\vec{c}|\cos\theta

-2 = 1 \times 2 \times \cos\theta

-2 = 2\cos\theta

\cos\theta = -1

When

\cos\theta = -1

\theta = 180°

(or

\pi

radians)

Therefore, the angle between vectors

\vec{b}

and

\vec{c}

180°