Let v1 =(0,4,7)andv2 =(3,2,1). Find 1. (i) length of v1. (ii) v1 + v2. (iii) len

Question

Let v1 =(0,4,7)andv2 =(3,2,1). Find

1. (i) length of v1.
(ii) v1 + v2.
(iii) length of v1 + v2.
(iv) v1 v2.
(v) length of v1 v2.
(vi) angle between v1 and v2. (vii) v1 · v2.
(viii) v1 × v2.

(2) Letv1 =(2,7,1)andv2 =(3,5,0). Find

(i) length of v1.
(vi) angle between v1 and v2. (vii) v1 · v2.
(viii) v1 × v2.

(3) Test if the following vectors are linearly independent. In2-Dspace,v1 =(0,3),v2 =(5,2).

(4) Test if the following vectors are linearly independent. In3-Dspace,v1 =(1,0,2),v2 =(2,1,2),v3 =(3,2,1).

(5) (20 pts) In 3-D space, let

v1 = (0, 2, 1),

v2 = (2, 1, 1),

v3 = (1, 3, 1),

v4 = (4, 2, 1).
Write v4 as the linear combination of v1, v2, v3.

Explanation / Answer

1 ( i) length of v1 = ||v1|| = sqrt ( 0^2 +4^2 + 7^2) = sqrt ( 16 +49 = sqrt60= 2sqrt15

(ii) v1 +v2 = ( 0, -4 , 7) +( 3, -2, -1) = ( 3 , -5 , 6)

(iii) lenght of (v1+v2) = sqrt ( 3^2 +5^2 +6^2) = sqrt ( 9+25 +36) =sqrt(70)

(iv) v1-v2 =  ( 0, -4 , 7) -( 3, -2, -1) = ( -3, -2, 8)

(v) length of (v1 -v2) = sqrt (3^2 + 2^2 +8^2) = sqrt(9+4+64 ) =sqrt77

(vi) Angle between v1 and v2

dot product of v1 and v2 = ( 0, -4 , 7)*( 3, -2, -1) = ( 0+ -4*-2 + 7*-1) = 8-7 =1

||v1|| = sqrt60 and ||v2|| = sqrt(3^2 +2^2 +1) = sqrt15

cosx = v1*v2/(||v1||*||v2|| = 1/(sqrt(60*15) = 1/30 = 0.033 radians

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