Question

If \[\vec a,\vec b,\vec c\] are vectors such that \[\vec a + \vec b + \vec c = 0\] and \[\left| {\vec a} \right| = 7,\left| {\vec b} \right| = 5,\left| {\vec c} \right| = 3\]then the angle between the vectors \[\vec b\] and \[\vec c\] is
A. \[{60^ \circ }\]
B. \[{40^ \circ }\]
C. \[{70^ \circ }\]
D. \[{20^ \circ }\]

Answer 1

Hint: In the given question, we need to find the value of the angle between the vectors \[\vec b\] and \[\vec c\]. For this, we will use the relation \[\vec a + \vec b + \vec c = 0\] and the definition of the dot product to get the desired result.


Formula used: The following formula is used for solving the given question.
The dot product of two vectors such as \[\vec u\] and \[\vec v\] is given by \[\vec u \cdot \vec v = uv\cos \theta \]

Complete step by step solution: We know that \[\vec a + \vec b + \vec c = 0\]
Here, \[\vec a,\vec b,\vec c\] are vectors.
Now, we will simplify \[\vec a + \vec b + \vec c = 0\]
\[\vec b + \vec c = - \vec a\]
By taking square on both sides, we get
\[{\left( {\vec b + \vec c} \right)^2} = {\left( { - \vec a} \right)^2}\]
By simplifying, we get
\[\left( {\vec b + \vec c} \right)\left( {\vec b + \vec c} \right) = \vec a \cdot \vec a\]
This gives
\[{\left| {\vec b} \right|^2} + {\left| {\vec c} \right|^2} + 2\vec b \cdot \vec c = {\left| {\vec a} \right|^2}\]
But \[\left| {\vec a} \right| = 7,\left| {\vec b} \right| = 5,\left| {\vec c} \right| = 3\]
Thus, we get
\[{\left( 5 \right)^2} + {\left( 3 \right)^2} + 2\vec b \cdot \vec c = {\left( 7 \right)^2}\]
\[2\vec b \cdot \vec c = 15\]
Also, \[2\left| {\vec b} \right|\left| {\vec c} \right|\cos \theta = 15\]
This gives,
\[30\cos \theta = 15\]
By simplifying, we get
\[\cos \theta = 1/2\]
\[\theta = {\cos ^{ - 1}}\left( {\dfrac{1}{2}} \right)\]
Thus, we get
\[\theta = {60^ \circ }\]
Hence, the angle between the vectors \[\vec b\] and \[\vec c\] is \[{60^ \circ }\] if If \[\vec a,\vec b,\vec c\] are vectors such that \[\vec a + \vec b + \vec c = 0\] and \[\left| {\vec a} \right| = 7,\left| {\vec b} \right| = 5,\left| {\vec c} \right| = 3\].


Thus, Option (A) is correct.

Additional Information: The definition of a vector is an entity with both magnitude and direction. The movement of an object between two points is described by a vector. The directed line segment can be used to geometrically represent vector mathematics. The magnitude of a vector is the length of the directed line segment, and the vector's direction is indicated by the angle at which it is inclined. A vector's "Tail" (the point where it begins) and "Head" (the point where it ends and has an arrow) are its respective names.

Note: Many students make mistake in simplification of vector expression such as \[\vec a + \vec b + \vec c = 0\]. This is the only way, through which we can solve the example in simplest way. Also, it is necessary to use the definition of dot product to get the desired result.