CONCEPT PREVIEW Refer to vectors a through h below. Make a copy or a sketch of each vector, and then draw a sketch to represent each of the following. For example, find a + e by placing a and e so that their initial points coincide. Then use the parallelogram rule to find the resultant, as shown in the figure on the right.





2c

Verified step by step guidance

1. Identify the vector labeled \( c \) from the given set of vectors. Make a clear sketch of this vector, noting its direction and magnitude.
2. Since the problem asks for \( 2c \), understand that this means scaling the vector \( c \) by a factor of 2. This involves doubling the length of vector \( c \) while keeping its direction the same.
3. On your sketch, draw the vector \( c \) starting from an initial point. Then, from the tip (head) of this vector, draw another vector identical in length and direction to \( c \).
4. The resultant vector \( 2c \) is the vector from the initial point of the first \( c \) to the tip of the second \( c \). This represents the sum \( c + c = 2c \).
5. Label the resultant vector \( 2c \) on your sketch to clearly show the doubled vector, and verify that its length is twice that of the original vector \( c \) and that it points in the same direction.

Key Concepts

Vector Addition

Vector addition involves combining two or more vectors to find a resultant vector. This is done by placing the initial point of one vector at the terminal point of another and then drawing the vector from the start of the first to the end of the last. It is essential for understanding how to combine vectors like a + e.

Parallelogram Rule

The parallelogram rule is a graphical method to add two vectors. By placing the vectors so their tails coincide, you complete a parallelogram with these vectors as adjacent sides. The diagonal of the parallelogram from the common tail point represents the resultant vector.

Scalar Multiplication of Vectors

Scalar multiplication involves multiplying a vector by a scalar (a real number), which changes the vector's magnitude without altering its direction unless the scalar is negative. For example, 2c means doubling the length of vector c while keeping its direction the same.