本研究探討高中生對於内稍概念中的基礎概念：「相同維度的任意兩向量皆可作内積」 之概念心像。研究採問卷調查法，收集質與量的資料。研究柚樣採立意取樣，包括高數學程 度、中數學程度學校之理組與文組四個班級，共149位高中三年級學生。研究結果發現高達 四成的學生對於起點不同的兩向量不具備皆可作内積的概念心像，僅有約四成的學生其任意 兩向量皆可作内積的心像穩固，不會隨著題目所給的刺激而有所改變；同時學生的概念心像 受到向量、角度、内積定義等相關概念心像的影響。

This study investigates the concept images of “any two vectors with same dimensions have an inner product" for senior high school students. A questionnaire survey was utilized to collect qualitative and quantitative data. A purposive sampling was used to sample 149 12th graders in four classes from a high or a middle level school with either science or liberal-arts oriented approach. The findings included that more than 40% students did not possess the concept images of having an inner product for vectors with different starting points; only about 40% students* concept images were stable that did not switch over different vector relations; and students' concept images in question were influenced by the related concept images of vectors, angles, and definition of inner products.