In Problem 3, the vector in the x -direction is 4 units...

In Problem $3,$ the vector in the $x$ -direction is 4 units long. Vectors 7 and $J$ are unit vectors in the $x$ - and $y$ -directions, respectively. A unit vector is a vector that is one unit long. So the $x$ -component of $\bar{p}$ is the vector $4 \vec{\imath},$ a product of a scalar and a vector. A scalar is a quantity that has only magnitude, not direction. Write the $y$ -component of $x$ in Figure $10-1 b$

In Problem $3,$ the vector in the $x$ -direction is
4 units long. Vectors 7 and $J$ are unit vectors in the $x$ - and $y$ -directions, respectively. A unit vector is a vector that is one unit long. So the $x$ -component of $\bar{p}$ is the vector $4 \vec{\imath},$ a product of a scalar and a vector. A scalar is a quantity that has only magnitude, not direction. Write the $y$ -component of $x$ in Figure $10-1 b$

Key Concepts

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar (a quantity with only magnitude) to change its size without affecting its direction. When a scalar is multiplied by a unit vector, it produces a vector in the same direction with a magnitude equal to the product of the scalar and the unit magnitude.

Basis Vectors

Basis vectors are a set of vectors that define the coordinate system in a vector space, typically chosen to be orthogonal and of unit length. In two-dimensional space, the standard basis vectors i and j represent the fundamental directions along the x- and y-axes, respectively, and any vector in the plane can be expressed as a linear combination of these basis vectors.

Unit Vectors

Unit vectors are vectors with a magnitude of one that indicate direction in a coordinate system. Commonly denoted as i (in the x-direction) and j (in the y-direction), they serve as the building blocks for expressing any vector as a sum of scaled unit vectors, with the scaling factors representing the magnitude in each respective direction.

Vector Components

A vector in a plane can be broken down into its horizontal and vertical (or x and y) components. This decomposition allows complex vectors to be analyzed and manipulated as independent parts, making it easier to understand and perform operations such as addition, subtraction, and scaling in each direction independently.

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