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What is a vector in the plane?
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A vector in the plane is a mathematical object with magnitude and direction, represented by an arrow from one point to another in the two-dimensional plane.
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How is a vector in the plane typically represented?
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A vector in a plane is typically represented by coordinates (x, y) which describe its position relative to the origin of a Cartesian coordinate system.
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What is the standard form of a vector in 2-dimensional space?
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The standard form of a vector in 2-dimensional space is expressed as v = ai + bj, where 'a' and 'b' are components of the vector along the x and y axes, respectively.
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How are vectors added together?
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Vectors are added together by adding their respective components: if v = (a, b) and u = (c, d), then v + u = (a+c, b+d).
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What is the difference between vectors and scalars?
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Vectors have both magnitude and direction, while scalars have only magnitude.
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What is the magnitude of a vector?
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The magnitude of a vector (a, b) is calculated as the square root of the sum of the squares of its components: √(a² + b²).
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How can you find the direction of a vector?
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The direction of a vector is often given by the angle it makes with the positive x-axis, calculated using the arctangent function: angle = atan2(b, a).
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What is a unit vector?
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A unit vector is a vector with a magnitude of 1, often used to indicate direction.
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How is a vector multiplied by a scalar?
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To multiply a vector by a scalar, multiply each component of the vector by the scalar: if v = (a, b) and k is a scalar, then kv = (ka, kb).
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What does it mean for two vectors to be equal?
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Two vectors are equal if they have the same magnitude and direction, meaning their corresponding components are equal.
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How can you determine if two vectors are parallel?
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Two vectors are parallel if they are scalar multiples of each other, meaning one vector can be expressed as a constant times the other.
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What is the dot product of two vectors and how is it calculated?
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The dot product of two vectors v = (a, b) and u = (c, d) is a scalar calculated as (a * c) + (b * d).
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What geometric information does the dot product provide?
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The dot product provides information about the angle between two vectors. A dot product of zero indicates that the vectors are perpendicular.
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What is the cross product of two vectors in the plane?
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In two-dimensional space, the cross product is not directly defined like in three dimensions, but the concept can be related by the magnitude of the determinant of a 2x2 matrix formed by their components.
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How is vector subtraction performed?
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Vector subtraction is performed by subtracting the corresponding components of the vectors: if v = (a, b) and u = (c, d), then v - u = (a-c, b-d).
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