
Find the summation of two vectors where the angle between the two vectors is .
Answer
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Hint : The angle between two vectors is considered the angle subtended from one to the other when the vectors are placed tail to tail. When the vectors are placed head to tail, the sum of the vector is another vector running from the tail of one to the head of the other.
Formula used: In this solution we will be using the following formula;
cosine rule: where is the magnitude of the sum of any vector, and are the individual vectors placed head to tail, and is the acute angle between them. where is the direction of the sum of any two vectors , and .
Complete step by step answer
Geometrically, the sum of a vector is given as shown above. When adding two vectors, the vectors are placed head to tail, then the sum is the vector drawn from the tail of the first vector to the head of the other. However, it can be proven that . The angle theta between them is considered the acute angle subtended if one is to orient one in the same direction as another, if the vectors are placed tail to tail. In the figure above, if we are to displace the to the left without changing the orientation until the tail coincides with the tail of , then the angle subtended by if it is to orient itself to face the direction of is considered the angle between two vectors. We must note that it’ll be equal to from the diagram.
Now, algebraically, from the cosine rule, the line is given as
But , then
Also,
Hence, by substitution
And this is the magnitude of the sum.
From trigonometry, the direction is given as
.
Note
To avoid confusions, let us proof that From a trigonometric identity,
,
Since, and
Hence, by substitution, we have that
Hence,
.
Formula used: In this solution we will be using the following formula;
cosine rule:
Complete step by step answer

Geometrically, the sum of a vector is given as shown above. When adding two vectors, the vectors are placed head to tail, then the sum is the vector drawn from the tail of the first vector to the head of the other. However, it can be proven that
Now, algebraically, from the cosine rule, the line
But
Also,
Hence, by substitution
And this is the magnitude of the sum.
From trigonometry, the direction is given as
Note
To avoid confusions, let us proof that
Since,
Hence, by substitution, we have that
Hence,
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