(a) Write each vector in Fig. E 1.37 in terms of the unit vectors and ĵ. (b) Use unit vectors t

step 1 Hi, here in this given problem first of all we draw a Cartesian plane having x axis y axis. In t

(a) Write each vector in Fig. E 1.37 in terms of the unit...

(a) Write each vector in Fig. $\mathrm{E} 1.37$ in terms of the unit vectors $\hat{\imath}$ and $\hat{j}$. (b) Use unit vectors to express vector $\overrightarrow{\boldsymbol{C}},$ where $\overrightarrow{\boldsymbol{C}}=3.00 \overrightarrow{\mathrm{A}}-4.00 \overrightarrow{\boldsymbol{B}} .$ (c) Find the magnitude and direction of $\overrightarrow{\boldsymbol{C}}$.

(a) Write each vector in Fig. $\mathrm{E} 1.37$ in terms of the unit vectors $\hat{\imath}$ and $\hat{j}$. (b) Use unit vectors to express vector $\overrightarrow{\boldsymbol{C}},$ where $\overrightarrow{\boldsymbol{C}}=3.00 \overrightarrow{\mathrm{A}}-4.00 \overrightarrow{\boldsymbol{B}} .$ (c) Find the
magnitude and direction of $\overrightarrow{\boldsymbol{C}}$.

Key Concepts

Vector Direction

Vector direction describes the angle a vector makes with a reference axis, typically the positive x-axis. It is usually determined using trigonometric functions, such as the arctangent function, which calculates the angle based on the ratio of the vertical to the horizontal components. Correctly determining the vector direction, while considering the vector’s quadrant, is important in order to accurately describe the vector’s orientation in space.

Vector Magnitude

The magnitude of a vector is a measure of its length and is computed using the Pythagorean theorem. When a vector is expressed in terms of its components, its magnitude is calculated by taking the square root of the sum of the squares of its components. Understanding how to compute vector magnitude is key in quantifying the size of a vector after various operations, such as addition or subtraction.

Scalar Multiplication and Vector Addition/Subtraction

Scalar multiplication involves multiplying each component of a vector by a real number, scaling its magnitude while preserving its direction (or reversing it if the scalar is negative). Vector addition and subtraction involve combining vectors by summing or subtracting their corresponding components. These operations are critical when constructing new vectors from existing ones, such as forming linear combinations or resolving net effects in physics problems.

Vector Components

Expressing a vector in terms of its components involves breaking it down into its horizontal (x) and vertical (y) parts. This process uses the standard unit vectors î and ?, where the vector is represented as a sum of its x-component multiplied by î and its y-component multiplied by ?. This decomposition is fundamental in several fields, including physics and engineering, as it allows for straightforward analysis and computation involving vectors.

Unit Vectors

Unit vectors are vectors with a magnitude of 1 that indicate direction. In a two-dimensional Cartesian coordinate system, the standard unit vectors, often denoted as î and ?, provide a basis from which any other vector can be expressed as a linear combination of these components. Using unit vectors simplifies the representation of vectors in a common coordinate system and is essential for vector decomposition.

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