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Practice: vectors, geometrically

Space geometry and vectors, grade 12 (4 units) Lesson 11 of 20Practice

Practice: vectors, geometrically

1

Simplify.

a

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

    b

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

      2

      uvABCKM

      In the triangle ABC\displaystyle ABC the medians meet at the point M\displaystyle M.
      AB→=u⃗\displaystyle {\overrightarrow{AB}}={\vec{u}}, AC→=v⃗\displaystyle {\overrightarrow{AC}}={\vec{v}}. Express in terms of u⃗\displaystyle \vec{u} and v⃗\displaystyle \vec{v}:

      a
      AM→\displaystyle \overrightarrow{AM}

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

        b
        CM→\displaystyle \overrightarrow{CM}

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

          3

          uvABCDMNO

          In the trapezoid ABCD\displaystyle ABCD (AD∥BC\displaystyle AD\parallel BC): AD=3BC\displaystyle {AD}={3BC}.
          The point O\displaystyle O is on the diagonal AC\displaystyle AC, and AO:OC=3:1\displaystyle {AO:OC}={3:1}.
          M\displaystyle M and N\displaystyle N are the midpoints of the sides BC\displaystyle BC and AD\displaystyle AD.
          AB→=u⃗\displaystyle {\overrightarrow{AB}}={\vec{u}}, BC→=v⃗\displaystyle {\overrightarrow{BC}}={\vec{v}}.

          a
          MN→\displaystyle \overrightarrow{MN}

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

            b
            MO→\displaystyle \overrightarrow{MO}

              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

              c

              Prove that M\displaystyle M, O\displaystyle O, N\displaystyle N are on one line.

              4

              uvwABCDSMN

              In the pyramid SABCD\displaystyle SABCD the base is the parallelogram ABCD\displaystyle ABCD.
              AB→=u⃗\displaystyle {\overrightarrow{AB}}={\vec{u}}, AD→=v⃗\displaystyle {\overrightarrow{AD}}={\vec{v}}, AS→=w⃗\displaystyle {\overrightarrow{AS}}={\vec{w}}.
              M\displaystyle M is the midpoint of SC\displaystyle SC, and N\displaystyle N the midpoint of SB\displaystyle SB.

              a
              AM→\displaystyle \overrightarrow{AM}

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                b
                NM→\displaystyle \overrightarrow{NM}

                  / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                  / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                  c

                  Is NM\displaystyle NM parallel to the base plane? Explain.

                  5

                  uvABCKEP

                  In the triangle ABC\displaystyle ABC: AB→=u⃗\displaystyle {\overrightarrow{AB}}={\vec{u}}, AC→=v⃗\displaystyle {\overrightarrow{AC}}={\vec{v}}.
                  K\displaystyle K is the midpoint of BC\displaystyle BC. The point E\displaystyle E is on AC\displaystyle AC, and AE:EC=1:2\displaystyle {AE:EC}={1:2}.
                  The segments AK\displaystyle AK and BE\displaystyle BE meet at the point P\displaystyle P.
                  Write AP→=t⋅AK→\displaystyle {\overrightarrow{AP}}={t\cdot\overrightarrow{AK}} and also AP→=AB→+s⋅BE→\displaystyle {\overrightarrow{AP}}={\overrightarrow{AB}+s\cdot\overrightarrow{BE}}, compare coefficients, and find the ratio AP:PK\displaystyle AP:PK.