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Given DA = a, AB = b, AB || DC, and DC = 2AB, express DB, DC, and BC in terms of a and b.

Mathematics

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Answers

3 parts
a

a+b\mathbf{a} + \mathbf{b}

b

2b2\mathbf{b}

c

−a+b-\mathbf{a} + \mathbf{b}

Step-by-step solution

7 steps

Here's how to solve this vector problem.

Given:
DA⃗=a\vec{DA} = \mathbf{a}
AB⃗=b\vec{AB} = \mathbf{b}
AB⃗\vec{AB} is parallel to DC⃗\vec{DC}
DC⃗=2AB⃗\vec{DC} = 2\vec{AB}

Part a· 2 steps

Express DB⃗\vec{DB} in terms of a\mathbf{a} and b\mathbf{b}.

  1. Use the triangle law of vector addition for △DAB\triangle DAB.

    To go from D to B, we can go from D to A, then from A to B.

    DB⃗=DA⃗+AB⃗\vec{DB} = \vec{DA} + \vec{AB}

  2. Substitute the given vector values.

    DB⃗=a+b\vec{DB} = \mathbf{a} + \mathbf{b}

    The vector DB⃗\vec{DB} is a+b\boxed{\mathbf{a} + \mathbf{b}}.

Part b

2b2\mathbf{b}

Express DC⃗\vec{DC} in terms of a\mathbf{a} and b\mathbf{b}.

  1. Use the given relationship between DC⃗\vec{DC} and AB⃗\vec{AB}.

    DC⃗=2AB⃗\vec{DC} = 2\vec{AB}

  2. Substitute the given vector value for AB⃗\vec{AB}.

    DC⃗=2b\vec{DC} = 2\mathbf{b}

    The vector DC⃗\vec{DC} is 2b\boxed{2\mathbf{b}}.

Part c

−a+b-\mathbf{a} + \mathbf{b}

Express BC⃗\vec{BC} in terms of a\mathbf{a} and b\mathbf{b}.

  1. Use the triangle law of vector addition for △DBC\triangle DBC.

    To go from B to C, we can go from B to D, then from D to C.

    BC⃗=BD⃗+DC⃗\vec{BC} = \vec{BD} + \vec{DC}

    Note that BD⃗=−DB⃗\vec{BD} = -\vec{DB}.

  2. Substitute the expressions for DB⃗\vec{DB} and DC⃗\vec{DC}.

    We found DB⃗=a+b\vec{DB} = \mathbf{a} + \mathbf{b} and DC⃗=2b\vec{DC} = 2\mathbf{b}.

    BC⃗=−(a+b)+2b\vec{BC} = -(\mathbf{a} + \mathbf{b}) + 2\mathbf{b}

  3. Simplify the expression.

    BC⃗=−a−b+2b\vec{BC} = -\mathbf{a} - \mathbf{b} + 2\mathbf{b} BC⃗=−a+b\vec{BC} = -\mathbf{a} + \mathbf{b}

    The vector BC⃗\vec{BC} is −a+b\boxed{-\mathbf{a} + \mathbf{b}}.

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