Contoh Perkalian Skalar

Nomor 1

Diketahui titik P (-3, -1, -v), Q(-ane, two, 0) dan R(1, 2, -2). Jika PQ = a dan QR + PR = b, maka a . b = …

A. 38

B. xxx

C. 26

D. 22

E. xvi

Pembahasan

Hitung terlebih dahulu QR, PR dan PQ

QR = R – Q = (one, 2, -2) – (-1, 2, 0) = (2, 0, -2)

PR = R – P = (1, 2, -2) – (-3, -one, -five) = (4, iii, three)

PQ = Q – P = (-1, two, 0) – (-3, -1, -5) = (2, 3, 5)

Hitung (QR + PR) = b

QR + PR = (2, 0, -2) + (four, 3, 3) = (6, 3, 1)

Jadi

a . b = PQ . (QR + PR) = (2, three, 5) . (6, 3, 1) = (2 . 6) + (3 . iii) + (v . 1) = 12 + ix + 5

a . b = 26

Jawaban: C

Nomor 2

Diketahui a = i – 3j + 2k, b = 2i + 8j + 2k, dan c = 8j + xk. Jika a . (b + c) = a . a maka nilai x adalah…

A. 42

B. 36

C.32

D. 28

E. 22

Pembahasan

a . (b + c) = a . a

a . b + a . c = a . a

Jadi sekarang hitung terlebih dahulu a . b, a . c, dan a . a

a . b = (i – 3j + 2k) . (2i + 8j + 2k) = (one . 2) + (-3 . 8) + (two . 2) = two – 24 + 4 = – xviii

a . c = (i – 3j + 2k) . (8j + xk) = (1 . 0) + (-3 . 8) + (2 . 10) = -24 + 2x

a . a =  (i – 3j + 2k) . (i – 3j + 2k) = (one . one) + (-3 . -3) + (2 . 2) = ane + 9 + 4 = 14

Jadi,

a . b + a . c = a . a

– 18 + (-24 + 2x) = 14

– 42 + 2x = xiv

2x = 56

ten = 56 / 2 = 28

Jawaban: D

Nomor 3

Jika |u| = 5, |v| = 3 dan |u + v| = vii, maka |u – v| =…

A. √19

B. five

C. √31

D. 6

E. √42

Pembahasan

Hitung terlebih dahulu sudut θ

|u + five| = seven

√(|u|
²

+ |v|
²

+ 2 . |u| . |v| . cos θ) = 7

√(five
^two

+ three
²

+ 2 . five . 3 . cos θ) = seven

 34 + 30 cos θ = 49

30 cos θ = 49 – 34 = 15

cos θ = 15/30 = one/2

θ = lx

Menghitung |u – five|

|u – v| = √(|u|
²

+ |v|
²

– 2 . |u| . |v| . cos θ)

|u – v| = √(v
^two

+ three
^two

– 2 . 5 . 3 . cos sixty)

|u – v| = √25 + nine – xxx . 1/2 = √34 – 15 = √19

Jawaban: A

Nomor 4

Diketahui:

Maka a . (b + c) = ….

A. 2

B. 5

C. 12

D. 25

E. xxx

Pembahasan

Hitung terlebih dahulu b + c

atau b + c = -4i + 3j + 0k

a = i + 2j + 3k

Menghitung a . (b + c)

a . (b + c) = (i + 2j + 3k) . (-4i + 3j + 0k) = (1 . -4) + (2 . three) + (3 . 0) = 2

Jawaban: A