向量运算与相关想法 [Math 241] - Section 12.2
# Arithmetric with vectors# add/ subtract/ scale vectors不同维度的向量无法相互加减乘除
# Geometric point of viewi.e.
# add
a ⃗ \vec {a} a b ⃗ \vec {b} b # scaling
# subtraction
# Algebratic point of viewi.e.a ⃗ = < a 1 , a 2 , . . . , a n > \vec {a} = <a_1, a_2, ..., a_n> a = < a 1 , a 2 , . . . , a n > b ⃗ = < b 1 , b 2 , . . . , b n > \vec {b} = <b_1, b_2, ..., b_n> b = < b 1 , b 2 , . . . , b n >
# adda ⃗ + b ⃗ = < a 1 + b 1 , a 2 + b 2 , . . . , a n + b n > \vec {a} + \vec {b} = <a_1 + b_1, a_2 + b_2, ..., a_n + b_n> a + b = < a 1 + b 1 , a 2 + b 2 , . . . , a n + b n >
# scalingc a ⃗ = < c a 1 , c a 2 , . . . , c a n > c \vec {a} = <c a_1, c a_2, ..., c a_n> c a = < c a 1 , c a 2 , . . . , c a n >
# subtractiona ⃗ − b ⃗ = < a 1 − b 1 , a 2 − b 2 , . . . , a n − b n > \vec {a} - \vec {b} = <a_1 - b_1, a_2 - b_2, ..., a_n - b_n> a − b = < a 1 − b 1 , a 2 − b 2 , . . . , a n − b n >
# Length / magnitudeGeometric - in R 2 \mathbb{R} ^ 2 R 2 R 3 \mathbb{R} ^ 3 R 3 a ⃗ = a 1 , a 2 \vec {a} = a_1, a_2 a = a 1 , a 2
Algebratic defination∣ a ⃗ ∣ = a 1 2 + a 2 2 + . . . + a n 2 |\vec {a}| = \sqrt {a_1 ^ 2 + a_2 ^ 2 + ... + a_n ^ 2} ∣ a ∣ = a 1 2 + a 2 2 + . . . + a n 2 ∣ a ⃗ ∣ |\vec {a}| ∣ a ∣ a ⃗ . \vec {a}. a .
vector of length 1< 3 5 , 4 5 > <\frac {3}{5}, \frac {4}{5}> < 5 3 , 5 4 > ( 3 5 ) 2 + ( 4 5 ) 2 = 1 \sqrt {(\frac {3}{5}) ^ 2 + (\frac {4}{5}) ^ 2} = 1 ( 5 3 ) 2 + ( 5 4 ) 2 = 1
# Unit Basic VectorsIn R 3 \mathbb{R} ^ 3 R 3
i ⃗ = < 1 , 0 , 0 > \vec {i} = <1, 0, 0> i = < 1 , 0 , 0 > j ⃗ = < 0 , 1 , 0 > \vec {j} = <0, 1, 0> j = < 0 , 1 , 0 > k ⃗ = < 0 , 0 , 1 > \vec {k} = <0, 0, 1> k = < 0 , 0 , 1 > # FactAny vector a ⃗ = < a 1 , a 2 , a 3 > \vec {a} = <a_1, a_2, a_3> a = < a 1 , a 2 , a 3 > R 3 \mathbb{R} ^ 3 R 3 i ⃗ , j ⃗ , k ⃗ \vec{i}, \vec{j}, \vec{k} i , j , k a ⃗ = i ⃗ ∗ a 1 + j ⃗ ∗ a 2 + k ⃗ ∗ a 3 \vec {a} = \vec {i} *a_1 + \vec {j}*a_2 + \vec {k} *a_3 a = i ∗ a 1 + j ∗ a 2 + k ∗ a 3
# BIG IDEA: LINEAR COMBINATIONlinear combination of vectors (in R n \mathbb{R} ^ n R n a 1 ⃗ , a 2 ⃗ , . . . , a k ⃗ \vec {a_1}, \vec {a_2}, ..., \vec {a_k} a 1 , a 2 , . . . , a k c 1 c_1 c 1 a 1 ⃗ \vec {a_1} a 1 c 2 c_2 c 2 a 2 ⃗ \vec {a_2} a 2 c k c_k c k a k ⃗ . \vec {a_k}. a k .
c 1 , c 2 , c k c_1, c_2, c_k c 1 , c 2 , c k linear combination of i ⃗ , j ⃗ , k ⃗ \vec {i}, \vec {j}, \vec {k} i , j , k # Above face restatedAny vectors in R 3 \mathbb{R} ^ 3 R 3 i ⃗ , j ⃗ , k ⃗ \vec {i}, \vec {j}, \vec {k} i , j , k < 2 , 4 , 1 > <2, 4, 1> < 2 , 4 , 1 > 2 i ⃗ + 4 j ⃗ + 1 k ⃗ . 2 \vec {i} + 4 \vec {j} + 1 \vec {k}. 2 i + 4 j + 1 k .
# Proof of facti.e.< a 1 , a 2 , a 3 > = a 1 i ⃗ + a 2 j ⃗ + a 3 k ⃗ . <a_1, a_2, a_3> = a_1\vec {i} + a_2\vec {j} + a_3\vec {k}. < a 1 , a 2 , a 3 > = a 1 i + a 2 j + a 3 k . a 1 i ⃗ + a 2 j ⃗ + a 3 k ⃗ . a_1 \vec {i} + a_2 \vec {j} + a_3 \vec {k}. a 1 i + a 2 j + a 3 k . a 1 < 1 , 0 , 0 > + a 2 < 0 , 1 , 0 > + a 3 < 0 , 0 , 1 > a_1<1, 0, 0> + a_2<0, 1, 0> + a_3<0, 0, 1> a 1 < 1 , 0 , 0 > + a 2 < 0 , 1 , 0 > + a 3 < 0 , 0 , 1 > i ⃗ , j ⃗ , k ⃗ . \vec {i}, \vec {j}, \vec {k}. i , j , k . < a 1 ∗ 1 , 0 , 0 > + < 0 , a 2 ∗ 1 , 0 > + < 0 , 0 , a 3 ∗ 1 > <a_1*1, 0, 0> + <0, a_2*1, 0> + <0, 0, a_3*1> < a 1 ∗ 1 , 0 , 0 > + < 0 , a 2 ∗ 1 , 0 > + < 0 , 0 , a 3 ∗ 1 > < a 1 + 0 + 0 , 0 + a 2 + 0 , 0 + 0 + a 3 > <a_1+0+0, 0+a_2+0, 0+0+a_3> < a 1 + 0 + 0 , 0 + a 2 + 0 , 0 + 0 + a 3 > < a 1 , a 2 , a 3 > <a_1, a_2, a_3> < a 1 , a 2 , a 3 >
# Pizzle of the day - 每日思考题For the Fact in Unit Basic Vectors before:< 1 , 0 , − 1 > , < 0 , 1 , 1 > , < 1 , 1 , 0 > <1, 0, -1>, <0, 1, 1>, <1, 1, 0> < 1 , 0 , − 1 > , < 0 , 1 , 1 > , < 1 , 1 , 0 > i ⃗ , j ⃗ , k ⃗ ? \vec {i}, \vec {j}, \vec {k}? i , j , k ?
