Textbook Question
In Exercises 13–20, let v be the vector from initial point P₁ to terminal point P₂. Write v in terms of i and j.P₁ = (-3, 4), P₂ = (6, 4)
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8. Vectors
Geometric Vectors
2:26 minutesProblem 24
Textbook Question
In Exercises 22–24, sketch each vector as a position vector and find its magnitude.v = -3j
Verified step by step guidance1
Step 1: Understand the vector notation. The vector \( \mathbf{v} = -3\mathbf{j} \) is given in terms of the unit vector \( \mathbf{j} \), which represents the direction along the y-axis.
Step 2: Sketch the vector. Since \( \mathbf{v} = -3\mathbf{j} \), this means the vector points in the negative y-direction with a magnitude of 3. On a coordinate plane, start at the origin (0,0) and draw a line 3 units downwards along the y-axis.
Step 3: Recall the formula for the magnitude of a vector. The magnitude of a vector \( \mathbf{v} = a\mathbf{i} + b\mathbf{j} \) is given by \( \|\mathbf{v}\| = \sqrt{a^2 + b^2} \).
Step 4: Apply the formula to find the magnitude. In this case, \( a = 0 \) and \( b = -3 \). Substitute these values into the formula: \( \|\mathbf{v}\| = \sqrt{0^2 + (-3)^2} \).
Step 5: Simplify the expression under the square root. Calculate \( (-3)^2 \) and then take the square root to find the magnitude of the vector.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Position Vectors
A position vector represents a point in space relative to an origin. In a two-dimensional Cartesian coordinate system, a position vector is expressed in terms of its components along the x and y axes. For example, the vector v = -3j indicates a point that is 3 units in the negative y-direction, with no displacement along the x-axis.
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Magnitude of a Vector
The magnitude of a vector is a measure of its length and is calculated using the Pythagorean theorem. For a vector expressed in component form as v = ai + bj, the magnitude is given by the formula |v| = √(a² + b²). In the case of v = -3j, the magnitude is simply the absolute value of the y-component, which is 3.
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Unit Vectors
A unit vector is a vector that has a magnitude of one and indicates direction. It is often used to represent the direction of a vector without regard to its magnitude. To convert a vector into a unit vector, you divide the vector by its magnitude. For the vector v = -3j, the unit vector would be -j, indicating the same direction but with a magnitude of 1.
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