向量的点乘积

两个向量的点乘积 是 向量a在向量b上的投影的长度 乘以 向量b的长度。
Another very important interpretation is:

\(\mathbf{a}\cdot\mathbf{b}=|\mathbf{a}||\mathbf{b}|\cos\theta\)

where:

• \((|\mathbf{a}|)\) = length of vector \((\mathbf{a})\)
• \((|\mathbf{b}|)\) = length of vector \((\mathbf{b})\)
• \((\theta)\) = angle between them

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The dot product (also called the scalar product) of two vectors measures how much the vectors point in the same direction.

For two vectors:

\[\mathbf{a} = (a_1, a_2, a_3) \]

\[\mathbf{b} = (b_1, b_2, b_3) \]

their dot product is:

\(\mathbf{a}\cdot\mathbf{b}=a_1b_1+a_2b_2+a_3b_3\)

向量的点乘可以写成向量的分量的乘积的和。

因为向量点乘的定义是 向量a在向量b上的投影的长度 乘以 向量b的长度 ，然后我们肉眼观察可得上面图中第二个等式。

这里我们将W向量继续拆成两个沿xy轴的分量，然后消掉垂直分量的乘积， 最后的公式就是 向量的分量的乘积的和。

也就是 VW = Vx*Wx + Vy*Wy

\(\mathbf{a}\cdot\mathbf{b}=|\mathbf{a}||\mathbf{b}|\cos\theta\) 最终能够写成 \(\mathbf{a}\cdot\mathbf{b}=a_xb_x+a_yb_y+a_zb_z\) 形式，一共有两个原因：

1. 向量a拆出来 \(a_x a_y a_z\)分向量首尾相接，是连续的，因此它们在向量b上的投影也是首尾相接的，并且最终的和等于 a在b上的投影， 证明了 向量的内积 满足分配律
2. 向量a拆出来的 分向量 \(a_x a_y a_z\) 互相垂直，这使得我们继续将b拆成\(b_x b_y b_z\)时，x y z互相垂直的向量内积为0，消除了很多项，最终只剩下同轴向量的点乘积

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Geometric meaning

The dot product tells you how aligned two vectors are.

1. Positive dot product

If:

\[\mathbf{a}\cdot\mathbf{b} > 0 \]

the vectors point roughly in the same direction.

Example: angle less than 90\(^{\circ}\)

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2. Zero dot product

If:

\[\mathbf{a}\cdot\mathbf{b} = 0 \]

the vectors are perpendicular (orthogonal).

Example: angle is 90\(^{\circ}\)

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3. Negative dot product

If:

\[\mathbf{a}\cdot\mathbf{b} < 0 \]

the vectors point in mostly opposite directions.

Example: angle greater than 90\(^{\circ}\)

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Projection intuition

You can think of the dot product as:

“How much of one vector goes in the direction of the other vector.”

If one vector points strongly along the other, the dot product becomes large.

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Example

Let:

\[\mathbf{a} = (3,4) \]

\[\mathbf{b} = (2,1) \]

Then:

\[\mathbf{a}\cdot\mathbf{b} = 3\times2 + 4\times1 = 10 \]

Since the result is positive, the vectors generally point in similar directions.

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In physics

The dot product appears everywhere.

For example, work is:

\(W=\mathbf{F}\cdot\mathbf{d}\)

Meaning:

• Only the part of the force in the direction of movement does work.
• A perpendicular force contributes nothing.

Example:

• carrying a bag horizontally:

  • your upward force is perpendicular to motion
  • dot product is zero
  • mechanical work on the bag is zero