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Geometric Vectors quiz #1

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  • Which transformation can map one triangle onto another using two reflections?
    A triangle can be mapped onto another congruent triangle using two reflections if the triangles are related by a translation or a rotation. Two reflections over parallel lines result in a translation, and two reflections over intersecting lines result in a rotation.
  • If a segment TQ is 26 units long, and QV is another segment related by vector addition or subtraction, how can you determine the length of QV?
    To determine the length of QV, use the vector addition or subtraction method. If QV is the result of adding or subtracting vectors with known magnitudes, apply the tip-to-tail method and calculate the magnitude accordingly. Without specific direction or relation, the length remains 26 units if QV is congruent to TQ.
  • What is one basic way to imply three-dimensional space in geometric representations?
    One basic way to imply three-dimensional space is by using vectors with three components (for example, in the form ai + bj + ck), which represent magnitude and direction in three dimensions.
  • If vector a is parallel to vector b and vector e is parallel to vector f, what does this imply about their directions?
    If vector a is parallel to vector b and vector e is parallel to vector f, this implies that each pair of vectors has the same or exactly opposite direction, differing only by a scalar multiple.
  • If MNOP is a square, what can be said about the vectors representing its sides?
    If MNOP is a square, the vectors representing its sides have equal magnitude and are orthogonal (perpendicular) to each other.
  • Which points are reflections of each other across both axes in a coordinate plane?
    Points that are reflections of each other across both axes have coordinates (x, y) and (−x, −y).
  • If point S lies between points R and T on a line segment and RT is 10 centimeters long, how can you express the position of S using vectors?
    The position of S can be expressed as a vector RS, where RS + ST = RT. If S divides RT into segments of lengths a and b such that a + b = 10, then the vector from R to S has magnitude a, and from S to T has magnitude b.
  • If AE is a chord of circle O and its length is 7 units, how can you represent this using vectors?
    The chord AE can be represented as a vector from point A to point E with magnitude 7 units.
  • If point A is the center of a circle and the ratio of the lengths of two radii is 2:1, what does this mean in terms of vectors?
    This means that one radius vector is twice as long as the other, or if the shorter radius is r, the longer radius is 2r.
  • What does it mean for points to lie on the same plane in terms of vectors?
    Points lie on the same plane if the vectors connecting them can be expressed as linear combinations of two non-parallel vectors in that plane.
  • What are orthogonal lines in the context of vectors?
    Orthogonal lines are lines whose direction vectors are perpendicular to each other, meaning their dot product is zero.
  • If vectors u and v have magnitudes |u| = |v| = 1 and u + v + w = 0, what is the magnitude of vector w?
    If u + v + w = 0, then w = −(u + v). The magnitude |w| depends on the angle between u and v: |w| = sqrt(|u + v|^2) = sqrt( (u + v) • (u + v) ). If u and v are unit vectors, |w| = sqrt(2 + 2cosθ), where θ is the angle between u and v.
  • If triangle XYZ is transformed by a dilation centered at O with scale factor 3, how does the length of side XY change?
    The length of side XY after the dilation will be three times its original length.
  • How do you find a unit vector in the same direction as the vector 4i − j + 8k?
    To find a unit vector in the same direction, divide the vector by its magnitude: The magnitude is sqrt(4^2 + (−1)^2 + 8^2) = sqrt(16 + 1 + 64) = sqrt(81) = 9. The unit vector is (4/9)i − (1/9)j + (8/9)k.