8.6.3 Vektor, SPM Praktis (Kertas 1)

January 23, 2022October 7, 2020 by

Soalan 5:

\begin{array}{l} p = 5 \underset{˜}{a} - 7 \underset{˜}{b} \\ q = - 2 \underset{˜}{a} + 3 \underset{˜}{b} \\ r = (h - 1) \underset{˜}{a} + (h + k) \underset{˜}{b}, \\ dengan keadaan h dan k adalah pemalar \end{array}

Gunakan maklumat di atas untuk mencari nilai h dan nilai k apabila r = 2p – 3q.

Penyelesaian:

\begin{array}{l} r = 2 p - 3 q \\ (h - 1) \underset{˜}{a} + (h + k) \underset{˜}{b} = 2 (5 \underset{˜}{a} - 7 \underset{˜}{b}) - 3 (- 2 \underset{˜}{a} + 3 \underset{˜}{b}) \\ (h - 1) \underset{˜}{a} + (h + k) \underset{˜}{b} = 10 \underset{˜}{a} - 14 \underset{˜}{b} + 6 \underset{˜}{a} - 9 \underset{˜}{b} \\ (h - 1) \underset{˜}{a} + (h + k) \underset{˜}{b} = 16 \underset{˜}{a} - 23 \underset{˜}{b} \end{array}

Bandingkan vektor,

h – 1 = 16

h = 17

h + k = –23

17 + k = –23

k = –40

Soalan 6:

Titik-titik P, Q dan R adalah segaris. Diberi bahawa

\vec{P Q} = 4 \underset{˜}{a} - 2 \underset{˜}{b} dan \vec{Q R} = 3 \underset{˜}{a} + (1 + k) \underset{˜}{b}

, dengan keadaan k ialah pemalar.
Cari

(a) nilai k,

(b) nisbah PQ : QR.

Penyelesaian:

(a)

\begin{array}{l} Jika P, Q dan R adalah segaris, \\ \vec{P Q} = m \vec{Q R} \\ 4 \underset{˜}{a} - 2 \underset{˜}{b} = m [3 \underset{˜}{a} + (1 + k) \underset{˜}{b}] \\ 4 \underset{˜}{a} - 2 \underset{˜}{b} = 3 m \underset{˜}{a} + m (1 + k) \underset{˜}{b} \\ Bandingan vektor: \\ \underset{˜}{a} : 4 = 3 m \\ m = \frac{4}{3} \\ \underset{˜}{b} : - 2 = m (1 + k) \\ - 2 = \frac{4}{3} (1 + k) \\ 1 + k = - \frac{6}{4} \\ k = - \frac{3}{2} - 1 \\ k = - \frac{5}{2} \end{array}

(b)

\begin{array}{l} \vec{P Q} = m \vec{Q R} \\ \vec{P Q} = \frac{4}{3} \vec{Q R} \\ \frac{\vec{P Q}}{\vec{Q R}} = \frac{4}{3} \\ ∴ P Q : Q R = 4 : 3 \end{array}

Soalan 7:

Diberi bahawa

\underset{˜}{x} = 3 \underset{˜}{i} + m \underset{˜}{j} dan \underset{˜}{y} = 4 \underset{˜}{i} - 3 \underset{˜}{j},

cari nilai m jika vektor

\underset{˜}{x} selari dengan vektor \underset{˜}{y} .

Penyelesaian:

\begin{array}{l} Jika vektor \underset{˜}{x} selari dengan vektor \underset{˜}{y} \\ \underset{˜}{x} = h \underset{˜}{y} \\ (3 \underset{˜}{i} + m \underset{˜}{j}) = h (4 \underset{˜}{i} - 3 \underset{˜}{j}) \\ 3 \underset{˜}{i} + m \underset{˜}{j} = 4 h \underset{˜}{i} - 3 h \underset{˜}{j} \\ Bandingkan vektor: \\ \underset{˜}{i} : 3 = 4 h \\ h = \frac{3}{4} \\ \underset{˜}{j} : m = - 3 h \\ m = - 3 (\frac{3}{4}) = - \frac{9}{4} \end{array}

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