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Vectors · Ages 17–18

Vector Proofs in Quadrilaterals Worksheet

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Designed for 12th grade math students. Express all segments in terms of the given base vectors, set up linear combinations, and equate coefficients to find unknown ratios.

  1. In parallelogram ABCD, vector AB = u and vector AD = v. Express the diagonal vector AC in terms of u and v.

  2. In parallelogram ABCD, vector AB = u and vector AD = v. Express the diagonal vector BD in terms of u and v.

  3. Point M is the midpoint of side BC in parallelogram ABCD with AB = u and AD = v. Express vector AM in terms of u and v.

  4. In trapezium ABCD, base AB is parallel to base DC, with DC = 2 * AB. Let vector AB = u and vector AD = v. Express diagonal vector AC in terms of u and v.

  5. Using the same trapezium ABCD (AB = u, AD = v, DC = 2u), express vector BC in terms of u and v.

  6. In trapezium ABCD with AB parallel to DC, M is the midpoint of AD and N is the midpoint of BC. Prove vector MN is parallel to the bases by expressing MN in terms of AB and DC.

  7. In parallelogram ABCD, point E lies on BC such that BE:EC = 1:2. Segment AE intersects diagonal BD at point P. Express vector AP in terms of u and v, and find the ratio in which P divides BD.

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Vector Proofs in Quadrilaterals Worksheet

Answer key

  1. In parallelogram ABCD, vector AB = u and vector AD = v. Express the diagonal vector AC in terms of u and v.

    u + v

  2. In parallelogram ABCD, vector AB = u and vector AD = v. Express the diagonal vector BD in terms of u and v.

    v - u

  3. Point M is the midpoint of side BC in parallelogram ABCD with AB = u and AD = v. Express vector AM in terms of u and v.

    u + (1/2)v

  4. In trapezium ABCD, base AB is parallel to base DC, with DC = 2 * AB. Let vector AB = u and vector AD = v. Express diagonal vector AC in terms of u and v.

    2u + v

  5. Using the same trapezium ABCD (AB = u, AD = v, DC = 2u), express vector BC in terms of u and v.

    u + v (calculated as -u + v + 2u)

  6. In trapezium ABCD with AB parallel to DC, M is the midpoint of AD and N is the midpoint of BC. Prove vector MN is parallel to the bases by expressing MN in terms of AB and DC.

    MN = (1/2)(AB + DC). Because MN is a scalar multiple of the base direction, it is parallel to both bases and equal to half their sum.

  7. In parallelogram ABCD, point E lies on BC such that BE:EC = 1:2. Segment AE intersects diagonal BD at point P. Express vector AP in terms of u and v, and find the ratio in which P divides BD.

    AP = (3/4)u + (1/4)v; point P divides BD in the ratio BP:PD = 1:3.