Draw each of the following vectors, label an angle that specifies the vector's direction, then find its magnitude and direction. r = (-2.0i - 1.0j) cm

검증된 단계별 안내

Step 1: Understand the vector notation. The vector r = (-2.0i - 1.0j) cm is expressed in terms of its components along the x-axis (i) and y-axis (j). Here, -2.0 cm is the x-component, and -1.0 cm is the y-component.

Step 2: Draw the vector on a Cartesian coordinate system. Start at the origin (0, 0). Move 2.0 cm to the left along the x-axis (negative direction) and then 1.0 cm downward along the y-axis (negative direction). Label the vector r and mark the angle it makes with the positive x-axis.

Step 3: Calculate the magnitude of the vector using the Pythagorean theorem. The magnitude |r| is given by the formula:

\sqrt{{2.0}^{2} + {1.0}^{2}}

. Substitute the values and simplify.

Step 4: Determine the direction of the vector. The angle θ (relative to the positive x-axis) can be found using the formula:

θ = \tan - 1 (\frac{-1.0}{-2.0})

. Note that both components are negative, so the vector lies in the third quadrant.

Step 5: Adjust the angle to reflect its position in the third quadrant. Since the arctangent function typically gives angles between -90° and 90°, add 180° to the result to find the correct angle in standard position. Express the magnitude and direction in the final form.

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주요 개념

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Vector Representation

Vectors are quantities that have both magnitude and direction, represented in a coordinate system. In this case, the vector r = (-2.0i - 1.0j) cm is expressed in terms of its components along the x-axis (i) and y-axis (j). The negative signs indicate that the vector points in the negative x and y directions, respectively.

Magnitude of a Vector

The magnitude of a vector is a measure of its length and can be calculated using the Pythagorean theorem. For the vector r = (-2.0i - 1.0j) cm, the magnitude is found by taking the square root of the sum of the squares of its components: |r| = √((-2.0)² + (-1.0)²) cm, which gives the total distance represented by the vector.

Direction of a Vector

The direction of a vector is often specified by an angle relative to a reference axis, typically the positive x-axis. This angle can be calculated using the arctangent function: θ = arctan(y/x). For the vector r = (-2.0i - 1.0j) cm, the angle will be in the third quadrant, reflecting its negative components, and can be expressed in degrees or radians.