Use vector operations to draw the resultant vector.
b) Draw $2 u -\frac{1}{3} v$.
Answer (1)
Multiply vector u by scalar 2 to get 2 u .
Multiply vector v by scalar 3 1 to get 3 1 v .
Subtract 3 1 v from 2 u to obtain the resultant vector 2 u − 3 1 v .
The resultant vector is 2 u − 3 1 v .
Explanation
Problem Analysis The problem asks us to perform vector operations to find the resultant vector of 2 u − 3 1 v . This involves scalar multiplication and vector subtraction. We will first multiply the vectors u and v by the scalars 2 and 3 1 respectively. Then, we will subtract the scaled vector 3 1 v from the scaled vector 2 u .
Scalar Multiplication of u by 2 First, we need to perform scalar multiplication on vector u by 2. This means we are scaling the vector u by a factor of 2, resulting in the vector 2 u .
Scalar Multiplication of v by 1/3 Next, we perform scalar multiplication on vector v by 3 1 . This means we are scaling the vector v by a factor of 3 1 , resulting in the vector 3 1 v .
Vector Subtraction Now, we need to subtract the vector 3 1 v from the vector 2 u . This is equivalent to adding the vector − 3 1 v to the vector 2 u . So, we have 2 u − 3 1 v = 2 u + ( − 3 1 v ) .
Resultant Vector The resultant vector is 2 u − 3 1 v . To draw this vector, we would start at the origin, move along the vector 2 u , and then move along the vector − 3 1 v . The final point reached is the endpoint of the resultant vector.
Examples
Vector operations are used extensively in physics to calculate resultant forces, velocities, and displacements. For example, if two forces are acting on an object, we can represent these forces as vectors and use vector addition to find the resultant force. Similarly, in navigation, vector operations are used to calculate the resultant displacement of a ship or aircraft, considering both its speed and direction.
