diff --git a/补课/多体动力学/05_vectors.ipynb b/补课/多体动力学/05_vectors.ipynb
index e69de29..2f83ab4 100644
--- a/补课/多体动力学/05_vectors.ipynb
+++ b/补课/多体动力学/05_vectors.ipynb
@@ -0,0 +1,92 @@
+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import sympy as sm\n",
+    "import sympy.physics.mechanics as me\n",
+    "sm.init_printing()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "q1, q2, q3, q4 = sm.symbols('q1:5')\n",
+    "l1, l2, l3, l4 = sm.symbols('l1:5')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "N, A, B, C = sm.symbols('N, A, B, C', cls=me.ReferenceFrame)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 8,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "A.orient_body_fixed(N, (q1, q2, 0), 'ZXZ')\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 9,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "B.orient_axis(A, q3, A.x)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 10,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "C.orient_body_fixed(B, (q3, q4, 0), 'XZX')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "r_P1_P2 = l1 * A.z\n",
+    "r_P1_P2"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "blade",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.16"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
diff --git a/补课/多体动力学/05_vectors.py b/补课/多体动力学/05_vectors.py
new file mode 100644
index 0000000..c0b5edd
--- /dev/null
+++ b/补课/多体动力学/05_vectors.py
@@ -0,0 +1,30 @@
+import sympy as sm
+import sympy.physics.mechanics as me
+sm.init_printing()
+
+q1, q2, q3, q4 = sm.symbols('q1:5')
+l1, l2, l3, l4 = sm.symbols('l1:5')
+
+N, A, B, C = sm.symbols('N, A, B, C', cls=me.ReferenceFrame)
+
+A.orient_body_fixed(N, (q1, q2, 0), 'ZXZ')
+print(A.dcm(N))
+
+B.orient_axis(A, q3, A.x)
+
+C.orient_body_fixed(B, (q3, q4, 0), 'XZX')
+
+r_P1_P2 = l1 * A.z
+print(r_P1_P2)
+
+r_P2_P3 = l2 * B.z
+print(r_P2_P3)
+
+r_P3_P4 = l3 *C.z - l4 *C.y
+print(r_P3_P4)
+
+r_P1_P4 = r_P1_P2 + r_P2_P3 + r_P3_P4
+print(r_P1_P4)
+
+print(r_P1_P4.express(B))
+print(r_P1_P4.free_symbols(N))
\ No newline at end of file
diff --git a/补课/多体动力学/06_Vector_Differentiation.md b/补课/多体动力学/06_Vector_Differentiation.md
new file mode 100644
index 0000000..e69de29
diff --git a/补课/多体动力学/sympy_start.ipynb b/补课/多体动力学/sympy_start.ipynb
index 6c2b697..e5e73c1 100644
--- a/补课/多体动力学/sympy_start.ipynb
+++ b/补课/多体动力学/sympy_start.ipynb
@@ -289,7 +289,7 @@
  ],
  "metadata": {
   "kernelspec": {
-   "display_name": "Turtle",
+   "display_name": "blade",
    "language": "python",
    "name": "python3"
   },
@@ -303,7 +303,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.11.8"
+   "version": "3.10.16"
   }
  },
  "nbformat": 4,